On Balloon Drawings of Rooted Trees

合集下载

高中英语课文原文和翻译分册txt版必修一选修六

THE BEST OF MANHATTAN’S ART GALLERIES
The Frick Collection (5th Avenue and E.70th Street)
Many art lovers would rather visit this small art gallery than any other in New York. Henry Clay Frick, a rich New Yorker, died in 1919, leaving his house, furniture and art collection to the American people. Frick had a preference for pre-twentieth century Western paintings, and these are well-represented in this excellent collection. You can also explore Frick's beautiful home and garden which are well worth a Visit.
The Impressionists were the first painters to work outdoors. They were eager to show how light and shadow fell on objects at different times of day. However, because natural light changes so quickly, the Impressionists had to paint quickly. Their paintings were not as detailed as those of earlier painters. At first, many people disliked this style of painting and became very angr about it. They said that the painters were careless and their paintings were ridiculous.

高考英语复习专题阅读：植树节题材阅读完型写作

高三复习专题阅读：植树节相关题材第一部分阅读理解(共8小题；每小题2.5分，满分20分)阅读下列短文，从每题所给的A、B、C、D四个选项中选出最佳选项。

AA rainforest is an area covered by tall trees with the total high rainfall spreading quite equally through the year and the temperature rarely dipping below l6℃. Rainforests have a great effect on the world environment because they can take in heat from the sun and adjust the climate. Without the forest cover, these areas would reflect more heat into the atmosphere, warming the rest of the world. Losing the rainforests may also influence wind and rainfall patterns, potentially causing certain natural disasters all over the world.In the past hundred years, humans have begun destroying rainforests in search of three major resources: land for crops, wood for paper and other products, land for raising farm animals. This action affects the environment as a whole. For example, a lot of carbon dioxide (二氧化碳) in the air comes from burning the rainforests. People obviously have a need for the resources we gain from cutting trees but we will suffer much more than we will benefit.There are two main reasons for this. Firstly, when people cut down trees, generally they can only use the land for a year or two. Secondly, cutting large sections of rainforests may provide a good supply of wood right now, but in the long run it actually reduces the world’s wood supply.Rainforests are often called the world’s drug store. More than 25% of the medicines we use today come from plants in rainforests. However, fewer than l% of rainforest plants have been examined for their medical value. It is extremely likely that our best chance to cure diseases lies somewhere in the world’s shrinking rainforests.1. Why can rainforests help to adjust the climate?A. They reflect more heat into the atmosphere.B. They bring about high rainfall throughout the world.C. They rarely cause the temperature to drop lower than l6℃.D. They reduce the effect of heat from the sun on the earth.2. What does the word “this”underlined in the third paragraph refer to?A. We will lose much more than we can gain.B. Humans have begun destroying rainforests.C. People have a strong desire for resources.D. Much carbon dioxide comes from burning rainforests.3. What can be inferred from the text ?A. We can get enough resources without rainforests.B. There is great medicine potential in rainforests.C. We will grow fewer kinds of crops in the gained land.D. The level of annual rainfall affects wind patterns.4. What might be the best title for the text?A. How to Save RainforestsB. How to Protect NatureC. Rainforests and the EnvironmentD. Rainforests and Medical DevelopmentBRainforests are home to a rich variety of medicinal plants, food, birds and animals. Can you believe that a single bush (灌木丛) in the Amazon may have more species of ants than the whole of Britain! About 480 varieties of trees may be found in just one hectare of rainforest.Rainforests are the lungs of the planet—storing vast quantities of carbon dioxide and producing a significant amount of the world’s oxygen. Rainforests have their own perfect system for ensuring their own survival; the tall trees make a canopy (树冠层) of branches and leaves which protect themselves, smaller plants, and the forest animals from heavy rain, intense dry heat from the sun and strong winds.Amazingly, the trees grow in such a way that their leaves and branches, although close together, never actually touch those of another tree. Scientists think this is the plants’way to prevent the spread of any tree diseases and make life more difficult for leaf-eating insects like caterpillars. To survive in the forest, animals must climb, jump or fly across the gaps. The ground floor of the forest is not all tangled leaves and bushes, like in films, but is actually fairly clear. It is where dead leaves turn into food for the trees and other forest life.They are not called rainforests for nothing! Rainforests can generate 75% of their own rain. At least 80 inches of rain a year is normal—and in some areas there may be as much as 430 inches of rain annually. This is real rain—your umbrella may protect you in a shower, but it won’t keep you dry if there is a full rainstorm. In just two hours, streams can rise ten to twenty feet. The humidity (湿气) of large rainforests contributes to the formation of rainclouds that may travel to other countries in need of rain.5. What can we learn about rainforests from the first paragraph?A. They produce oxygen.B. They cover a vast area.C.They are well managed.D. They are rich in wildlife.6. Which of the following contributes most to the survival of rainforests?A. Heavy rains.B. Big trees.C. Small plants.D. Forest animals.7.Why do the leaves and branches of different trees avoid touching each other?A. For more sunlight.B. For more growing space.C. For self-protection.D. For the detection of insects.8.What can be a suitable title for the text?A. Life-Giving RainforestsB. The Law of JungleC. Animals in the AmazonD. Weather in Rainforests第二部分完形填空(共15小题；每小题1分，满分15分)阅读下面短文，从每题所给的A、B、C、D四个选项中选出可以填入空白处的最佳选项。

英语大作文画桉树

英语大作文画桉树The Eucalyptus TreeThe eucalyptus tree is a remarkable and iconic plant native to Australia that has captivated the attention of people around the world. These tall, slender trees with their distinctive peeling bark and aromatic leaves have become a symbol of the Australian landscape and have found their way into the hearts and minds of many. Whether you are admiring the graceful silhouette of a eucalyptus tree against the backdrop of a setting sun or enjoying the refreshing scent of its essential oils, there is no denying the allure and significance of this remarkable plant.One of the most striking features of the eucalyptus tree is its height and stature. These trees can grow to towering heights, with some species reaching over 300 feet tall. This impressive size is a testament to the tree's resilience and adaptability, as it has evolved to thrive in the harsh and often inhospitable environments of Australia. The tall, slender trunks of the eucalyptus tree are covered in a distinctive peeling bark that adds to the tree's unique and captivatingappearance.Another fascinating aspect of the eucalyptus tree is its remarkable ability to adapt to a wide range of environmental conditions. These trees are found in a variety of habitats, from the arid deserts of central Australia to the lush, humid rainforests of the east coast. This adaptability is due in large part to the tree's unique physiology, which includes features such as waxy leaves that help to reduce water loss and a deep, extensive root system that allows the tree to access water and nutrients even in the most challenging of environments.One of the most well-known and widely-used products of the eucalyptus tree is its essential oils. These oils, which are extracted from the tree's leaves, have a wide range of medicinal and therapeutic applications. The active compounds in eucalyptus oil, such as cineole and citronellal, have been shown to have anti-inflammatory, antimicrobial, and decongestant properties, making them useful in the treatment of a variety of respiratory and skin conditions.In addition to its medicinal applications, eucalyptus oil is also widely used in the production of a variety of consumer products, from cleaning supplies to personal care items. The distinctive, refreshing scent of eucalyptus oil has made it a popular ingredient in everythingfrom household cleaners to natural perfumes and fragrances.Beyond its practical uses, the eucalyptus tree also holds significant cultural and symbolic significance in Australia and beyond. In Aboriginal culture, the eucalyptus tree is revered for its many uses, from the use of its wood for tools and shelter to the use of its leaves in traditional medicine and ceremony. The tree is also a prominent feature in Australian art and literature, with its distinctive silhouette and vibrant foliage appearing in countless paintings, photographs, and works of fiction.The eucalyptus tree's significance extends beyond Australia as well. In many parts of the world, the tree has been introduced as an ornamental and landscaping plant, and its rapid growth and adaptability have made it a popular choice for reforestation and land reclamation projects. In some regions, the eucalyptus tree has even become a naturalized species, with populations thriving in areas far from its native Australian habitat.Despite its many benefits and widespread popularity, the eucalyptus tree is not without its challenges. In some areas, the tree's rapid growth and ability to spread quickly has led to concerns about its impact on local ecosystems, with some species being considered invasive. Additionally, the tree's flammable leaves and bark have made it a potential fire hazard in some regions, leading to concernsabout its use in urban and suburban areas.Despite these challenges, the eucalyptus tree remains a beloved and iconic plant that continues to captivate people around the world. Whether you are admiring its towering height, enjoying the refreshing scent of its essential oils, or simply marveling at its resilience and adaptability, there is no denying the enduring appeal and significance of this remarkable tree.。

技能训练3U1听力原文

Section A1. W：I suppose you’ve bought some gifts for your family.M：Well，I’ve bought a shirt for my father, and two books for my sister. But I haven’t decided what to buy for my mother, probably some jewels.Q：Who did the man buy the books for?2. W：Look, it says they want a junior sales manager and it seems like it’s a big company. That’ll be good. So you might have to travel a lot.M：Do they say anything about the experience?Q：What are they talking about?3. W：I think we’ve covered every thing. What about a cup of coffee before we move on to the nextitem?M：Good idea. I really can’t wait another minute.Q：What does the woman suggest doing?4. W：But what happens if it rains? What are we going to do then?M：We’ll have to count on good weather. But if it does rain, the whole thing will have to be canceled.Q：What do we learn from the conversation?5. W：You took an optional course this semester, didn’t you? How is it going?M：Terrible. It seems like the more the professor talks, the less I understand.Q：How does the man feel about the course?6. W：Mark is playing computer games.M：Should he do that when the final exam is drawing near?Q：What does the man think Mark should do?7. M：Jack seems to thin k this year’s basketball season will be disappointing.W：That’s his opinion. Most others think differently.Q：What does the woman mean?8. M：Is this the check-in counter for flight 914 to Los Angeles?W：Yes. B ut I’m sorry the flight is delaye d because of a minor mechanical problem. Please wait for further notice. .Q：What do we learn from this conversation?9. M：Excuse me. I’d like to place an advertisement for a used car in this Sunday edition of your paper.W：Ok. B ut you have to run your advertisement all week. We can’t quote rates for just Sunday. Q：Where is the conversation most probably taking place?10. M：I spend so much time polishing my letter application.W：It’s worthwhile to make the effort.You know just how important it is to give a good impression.Q：What do we know about the man?Section BPassage OneNot everybody reads the daily newspaper. People who don’t read newspaper are sometimesreferred to as non-readers. Early research has shown that the non-readers are generally low in education, low in income, either very young or very old. In addition, non-readers are more likely to live in rural areas and have less contact with neighbors and friends. Other studies show that non-readers tend to isolate themselves from the community, and less likely to own a home and seldom belong to local voluntary organizations.Why don’t these people read daily paper? They say they don’t have the time; they prefer radio or TV; they have no interest in reading at all. And besides, they think newspapers are too expensive. Recent surveys, however, have indicated that the portrait of the non-reader is more complicated than first thought. There appears to be a group of non-readers that do not fit the type mentioned above. They are high in income and fall into the age group of 26 to 65. They are far more likely to report that they d on’t have the time to read the papers and they have no interest in the content. Editors and publishers are attempting to win them back. First, they are adding news briefs and comprehensive indexes. This will help overcome the time problem. And they are also giving variety to newspaper content to help build the readers’ interest.Questions 11 to 13 are based on the passage you have just heard.11. What is typical of non-readers according to the early research?12. What are the findings of recent surveys?13. What are editors and publishers doing to attract the non-readers?Passage TwoDid you know that there’s a kind of bird that can sew? This bird, called the tailor bird, uses its mouth as a needle. It sews leaves together in the shape of a cup. Then it adds a layer of straw to the inside of the cup and lays its eggs there. Each bird species builds its own special kind of nest. The most common materials used for nests are grasses, branches and feathers. A bird must weave these materials into a nest. Just imagine building a house without cement or nails to hold together. Another bird is called the weaver bird. The weaver bird builds a nest that looks like a basket. The nest is shaped like a pear with a hole in the middle. The hole is the door of the nest. A third bird is called the ovenbird. The ovenbird makes a nest that is very solid. The nest is made of mud. The ovenbird forms the mud into the shape of an oven and then let it dry in the sun. The sun bakes the mud making it very hard. Not all birds make their homes in branches. Some birds build their nests on the ground while others bury their eggs under the ground. And some birds do not build nests at all. So when you look for nests and eggs in branches of the trees and bushes, remember that some nests may be right under your feet.Questions 14 to 17 are based on the passage you have just heard.14. What does the nest built by tailor bird look like?15. Why is there a hole in the weaver bird’s nest?16. What is the oven bird’s nes t made of ?17. What might surprise us about birds’ nests according to the speaker?Passage ThreeYou can tell the age of a tree by counting its rings. B ut these records of trees’ life really say a lot more. Scientists are using tree rings to l earn what’s being happening on the sun’s surface for the last 10,000 years. Each ring represents a year of growth. As the tree grows, it adds a layer to itstrunk taking up chemical elements from the air. By looking up the elements in the rings for a given year, scientists can tell what elements were in the air that year. Doctors Stevenson is analyzing one element—carbon-14 in rings from both living and dead trees. Some of the rings go back almost 10,000 years to the end of the Ice Age. When Stevenson followed the carbon-14 trail back in time, he found carbon-14 levels change with the intensity of solar burning. You see, the sun has cycles. Sometimes it burns fiercely.At other times it’s relatively calm. During the sun’s violent periods, it throws off charged particles in fast moving strings called solar winds. The particles interfere with the formation of carbon-14 on earth. When th ere’s more solar wind activity less carbon-14 is produced. 10,000 years of tree rings show that the carbon-14 level rises and falls about every 420 years. The scientists concluded that the solar wind activity must follow the same cycle. Questions 18 to 20 are based on the passage you have just heard.18. What is the purpose of the scientists in studying tree rings?19. What affects the amount of carbon-14 on earth?20. What do we learn from the passage about the solar wind activity?Passage Two【短文大意】本文主要介绍了一种用嘴把树叶缝到一块儿的鸟如何编制自己的巢。

Unit1 Book 3(丛岩)

Unit 1Part ⅠListening ComprehensionA．答案与详解Section A1. 【答案与详解】D。

事实状况题。

对话中的细节辨认部分是“two books for my sister”，说给姐姐买了两本书，所以选D。

给父亲买的是衬衫，A错；为母亲买什么还未确定，故排除B；C对话中未提及。

2. 【答案与详解】A。

谈论话题题。

由junior sales manager, company以及experience等特定场境细节可以推出，说话双方谈论关于求职的情况，所以选A。

B错误表述职务，对话中是junior sales manager；C与对话中a big company矛盾；D对话中未提及。

3. 【答案与详解】A。

行为活动题。

对话中“What about of a cup of coffee”与“having a break”是同义转述，a cup of coffee在英美国家指的是coffee break，喝咖啡的休息时间，所以选A。

4. 【答案与详解】C。

行为活动题。

如果下雨，全部计划都得取消，C是这一句的同义转述，所以选C。

对话中“count on”是“指望”的意思，选项中count in是指“把…算在内”，故排除B；D项中postpone“推迟”，与对话中cancel不一致。

5. 【答案与详解】B。

观点态度题。

对话中“the more…the less…”表示越听越不懂，与B中“hard to follow”同义转述，所以选B。

6. 【答案与详解】C。

观点态度题。

“Should he…?”通过语气判断表示责备，大意是：快考试了他还该那样做吗？由常识可知，临考前应抓紧复习功课，所以选C。

7. 【答案与详解】A。

观点态度题。

从his重读推断女士对此不以为然，再从后面的differently判断女士与男士观点不同，所以选A。

8. 【答案与详解】D。

事实状况题。

对话中说：请等候通知，说明起飞时间未确定，与选项D同义转述。

教室里下了一张干净的图画英语作文5句

教室里下了一张干净的图画英语作文5句全文共6篇示例，供读者参考篇1There was a brand new whiteboard in our classroom today! I was so excited when I saw it. Our old whiteboard had gotten really yucky and stained over the years. Whenever you tried to write on it, the marker streaked and made a mess. Some parts were basically brown instead of white from all the built-up marker residue. It was so gross!This new whiteboard is pristine and beautiful. The surface is a perfect crisp white that just gleams. When you run your hand across it, it feels smooth as can be. No bumps, no scratches, no stains at all. It's like a fresh clean canvas just waiting for new ideas and lessons to come alive on it.I could hardly wait for class to start so we could begin using the new whiteboard. I just knew the marker would glide across it effortlessly, letting the ink flow out in perfect crisp lines. And I was right! When Mrs. Jones started writing out the agenda for the day, it looked so clear and neat. The letters and numbers formed beautifully with no skipping or blotching at all.During the math lesson, Mrs. Jones worked out a few example problems showing how to borrow and regroup for subtraction. With the old board, her writing always ended up looking cramped and messy. But today, each step was beautifully spaced out and easy to follow. Even the kids sitting in the back row could clearly see what she was doing. No more squinting and straining to make out the sloppy numbers!At one point, Jimmy accidentally dropped his marker and it went rolling across the floor. We all gasped, afraid it might leave a big ugly streak across the pristine new surface. But Mrs. Jones calmly picked it up and, wouldn't you know, not a single mark was left behind! She did have to wipe off a little dust with her cloth, but then the board looked as good as new again.I just loved watching Mrs. Jones draw diagrams and illustrations during our science lesson about the life cycle of plants. On the old board, her pictures always ended up looking blobby since the markers were so low-quality. But today, with the new smooth surface and good markers, her drawings looked like they could have come right out of a book. The vibrant green outlines of the leaves and stems looked so real andthree-dimensional.My favorite part was during our creative writing exercise at the end of the day. Mrs. Jones gave us each a square of the board to practice our descriptive skills by drawing something from our imagination and labeling the parts with descriptive words. I decided to draw the most colorful, fanciful hot air balloon I could think of. I used the red and orange markers to make the bright and vibrant balloon itself. The blue made a perfect sky backdrop with little puffy white clouds. I drew intricately designed baskets underneath made of yellow and brown woven rattan. I added a little green trailer attached to carry supplies and luggage. I worked really hard on all the tiny details like the burner canisters and instrument panels in the baskets.When I was done, my picture looked so amazing on that clean white surface. The篇2The Clean Picture in the ClassroomIn my classroom, something very exciting happened today! When we entered the classroom this morning, we found a clean and beautiful picture on the board. It was like a magical surprise waiting for us.The picture was full of vibrant colors, and it showed a lovely countryside scene. There were green fields with flowers, a shining sun in the blue sky, and a sparkling river flowing through the middle. It was so pretty that it made me smile instantly.All of us in the class were curious about who could have drawn such a wonderful picture. Our teacher asked each of us if we knew anything about it, but nobody had any idea. It was like a mystery waiting to be solved!We decided to investigate and become detectives for the day. We searched for clues around the classroom, looking for any hints that could lead us to the artist. We checked our desks, the pencil case, and even the bookshelves, but there was nothing that could help us solve the mystery.Next, we decided to ask the other students in different classes if they had seen anyone near our classroom during recess. We went to the playground and started talking to our friends from other classes. Some of them said they saw a girl with a sketchbook near our classroom, but they didn't know who she was.With this important clue, we went back to our classroom and started thinking about who the mysterious artist could be. We remembered that there was a new student who joined our classyesterday. Her name was Emily, and she loved drawing. Maybe she was the one who drew the picture!We rushed to find Emily and asked her if she was the artist behind the beautiful picture. She blushed and nodded shyly. Emily explained that she wanted to surprise us all with something nice, so she came early to school and drew the picture before anyone arrived.We were amazed by her talent and grateful for the surprise she prepared for us. Emily's picture made our classroom feel more cheerful and welcoming. We decided to keep it on the board for everyone to enjoy.Today was a special day for us. It taught us that sometimes, even the simplest things can bring joy and excitement to our lives. We also learned the importance of appreciating the efforts and talents of others. Emily's clean picture will always remind us of the power of creativity and friendship in our classroom.I can't wait to see what surprises await us in the future, but for now, I'll cherish the memory of the clean picture in our classroom.篇3A Fresh Drawing Falls in the ClassroomWow! Something really amazing happened in my classroom today. I was just sitting at my desk, minding my own business and trying to pay attention to Mrs. Johnson's math lesson on multiplication. I know times tables are super important, but they can be pretty tricky to memorize sometimes.Anyways, I was looking down at my worksheet, chewing on the end of my pencil and furrowing my brow as I tried to figure out what 7 x 8 equaled. Math is not my strongest subject for sure. That's when I heard a weird sort of fluttering noise above me. I glanced up and could hardly believe my eyes!A large piece of paper was drifting down from the ceiling, turning lazily in the air as it fell. As it twirled and twisted, I could see there was a beautiful drawing on it in bright colors. It almost looked like a stained glass window from an old church with the way the different hues blended together.I watched in awe, my mouth hanging open dumbly, as the paper kept spiraling down, down, down until it finally landed right on top of my math worksheet. The colors seemed to shimmer and glow, and I couldn't help but reach out to touch the mysterious drawing."Nathan! What on earth are you doing?" Mrs. Johnson's sharp voice broke through my trance-like state. I quickly pulled my hand back, flushing with embarrassment as I realized the whole class had gone silent and everyone was staring at me."I...uh...this drawing just fell from the ceiling, Mrs. Johnson," I stammered out, gesturing at the paper lying on my desk.To my surprise, instead of scolding me, Mrs. Johnson's eyes went wide with wonder. "A drawing? From the ceiling you say? How peculiar..." She scurried over to my desk, the other students craning their necks to try and get a better look.Mrs. Johnson gasped softly when she saw the image up close. "Why...it's spectacular! I've never seen anything like it."She was right - none of us had. The colors didn't look like anything you could get from a regular box of crayons or colored pencils. There were shades of amber and sapphire, crimson and viridian that seemed to almost glow from within. Swirling together, the hues created an incredibly detailed picture of a meadow filled with fanciful plants and flowers unlike any I'd ever seen before."Where did this come from?" Mrs. Johnson murmured under her breath, carefully picking up the large piece of paper to get a closer look.I could only shake my head slowly. "It just...fell from up there." I pointed up at the ceiling tiles overhead, suddenly noticing a slight distortion in the air right above where I was sitting. It looked almost like heat waves rising up from a highway on a scorching summer day.Before I could say anything else, the shimmering distortion grew larger and another piece of paper came fluttering down, this one with an intricately detailed drawing of a rainforest scene. Then another and another followed, drifting down in a steady stream to land all over the classroom.The other students began shouting with excitement, jumping up from their desks to catch the falling drawings and examine the breathtaking images up close. Mrs. Johnson tried in vain to regain control, looking hilariously flustered.For once, I didn't care about getting in trouble or being scolded to sit back down. This was the most incredible thing I'd ever witnessed! More drawings continued to appear from that odd ripple in the air above my desk - pictures of animals that looked like they came straight out of a fantasy world, strangealien landscapes, even a few jaw-dropping portraits that seemed more lifelike than any photograph.By the time the strange phenomenon finally ended, there must have been at least a hundred of those wondrous drawings scattered all around the classroom. My friends and I pored over them for the rest of the day, our earlier lessons entirely forgotten in our awe and amazement.I've never figured out where those incredible pictures came from or why they chose to materialize in our classroom. Part of me doesn't really want to know - some mysteries are better left unsolved. All I can say is that I'll never forget the day a universe of篇4In our classroom today, something amazing happened! As soon as we entered the classroom, we noticed there was a clean, blank canvas hanging on the wall. It was like a magical invitation for us to fill it with our imagination!Excitement filled the air as we gathered around the canvas, eagerly discussing what we should draw. Some of us suggested drawing a beautiful garden with colorful flowers, while otherswanted to create a scene from our favorite fairytale. We were all bubbling with ideas!Finally, we decided to draw a magnificent underwater world. With blue and green markers in hand, we started to sketch the outline of the ocean floor, complete with swaying seaweed and vibrant coral reefs. We added colorful fish, graceful dolphins, and even a curious octopus.As we worked together, our classroom transformed into a mini art studio. Each of us contributed to the masterpiece, carefully coloring and shading different parts of the picture. We giggled and laughed as we brought our imaginations to life on the canvas.Time flew by, and before we knew it, our artwork was complete. We stepped back to admire our creation—a breathtaking underwater world that seemed to come alive. The classroom was filled with a sense of pride and accomplishment.Our teacher, Mrs. Johnson, walked into the classroom and gasped in amazement. She was thrilled to see our collaborative effort and praised our creativity. She told us how proud she was of each and every one of us.With beaming smiles, we shared stories about the different elements we had drawn. Some of us talked about the fish we imagined swimming freely, while others described the adventures of the dolphins. It was a moment of pure joy and friendship.Our classroom now feels even more vibrant and alive with the beautiful picture adorning the wall. Every day, as we enter the classroom, we are reminded of the power of teamwork, imagination, and creativity. It serves as a constant inspiration for us to dream big and believe in ourselves.We are grateful for the opportunity to create art together and for the teachers who encourage us to express ourselves. Our classroom is not just a place for learning, but also a space where our dreams take flight through the strokes of our markers.In conclusion, the clean canvas in our classroom turned into a magical masterpiece that celebrates our unity and creativity. It reminds us that with a little imagination and teamwork, we can create something truly extraordinary. Our classroom has become a gallery of dreams, and we can't wait to see what other masterpieces we'll create in the future!篇5One sunny morning, when I entered the classroom, I saw a clean and beautiful picture on the blackboard. It was like a magical surprise waiting for us!（一个阳光明媚的早晨，当我走进教室时，看到黑板上画着一幅干净而美丽的图画。

托福TPO1阅读答案与翻译解析3

参考译文：山上树带界线的植被The transition from forest to treeless tundra on a mountain slope is often a dramatic one. Within a vertical distance of just a few tens of meters, trees disappear as a life-form and are replaced by low shrubs, herbs, and grasses. This rapid zone of transition is called the upper timberline or tree line. In many semiarid areas there is also a lower timberline where the forest passes into steppe or desert at its lower edge, usually because of a lack of moisture.通常从山坡上的森林到没有树的苔原是一种非常戏剧化的转变。

在一个垂直距离只有几十米的地方，树木这种生命形式就消失了，取而代之的是低矮的灌木、药草和牧草。

这种急速转变的区域被称为上行树带界线或林木线。

在许多干旱的地区存在着下行树带界线，在这里由于缺乏水分森林变成干草原，甚至在最下端会出现沙漠。

The upper timberline, like the snow line, is highest in the tropics and lowest in the Polar Regions. It ranges from sea level in the Polar Regions to 4,500 meters in the dry subtropics and 3,500-4,500 meters in the moist tropics. Timberline trees are normally evergreens, suggesting that these have some advantage over deciduous trees (those that lose their leaves) in the extreme environments of the upper timberline. There are some areas, however, where broadleaf deciduous trees form the timberline. Species of birch, for example, may occur at the timberline in parts of the Himalayas.上行树带界线，比如雪线，在热带最高在极地最低。

英译汉。高英

Lesson 11)The one I am thinking of particularly is entered by Gothicarched gateway of aged brick and stone .You pass from the heat and glare of a big, open square into a cool, dark cavern which extends as far as the eye can see, losing itself in the shadowy distance.此时此刻我要说的集市的入口是一座古老的砖石结构的哥特式拱门，年代非常久远。

当你穿过一个烈日暴晒的大型露天广场，然后走进一个凉爽、幽暗的洞穴。

这洞穴一直一直向前延伸，一眼望不到尽头，最终消失在远处的阴影里。

2)It is a point of honor with the customers not to let the shopkeeper guess what it is she really likes and want until the last moment. 对于顾客来说，至关重要的一点是，不到最后一刻是不能让店主猜到她心里究竟中意哪样东西、想买哪样东西的.3)The seller ,on the other hand ,makes a point of protesting that the price he is charging is depriving him of all profit, and that he is sacrificing this because of his personal regard for the customer. 而在店主那一方来说，则是竭尽全力的让顾客相信，他开出的价钱使他根本无利可图，而他之所以愿意这样做完全是出于本人对顾客的敬重。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

Journal of Graph Algorithms and Applications/vol.11,no.2,pp.431–452(2007)On Balloon Drawings of Rooted TreesChun-Cheng Lin1Hsu-Chun Yen1,2,∗1Department of Electrical EngineeringNational Taiwan University,Taipei,Taiwan106,ROCsanlin@.tw yen@.tw2Department of Computer ScienceKainan University,Taoyuan,Taiwan338,ROCAbstractAmong various styles of tree drawing reported in the literature,bal-loon drawing enjoys a desirable feature of displaying tree structures in arather balanced fashion.Each subtree in the balloon drawing of a tree isenclosed in a circle.Along any path from the root node,the radius of eachcircle reﬂects the number of descendants associated with the root node ofthe subtree.In this paper,we investigate various issues related to balloondrawings of rooted trees from the algorithmic viewpoint.First,we designan eﬃcient algorithm to optimize the angular resolution and the aspectratio for the balloon drawings of rooted unordered trees.For the case ofordered trees for which the center of the enclosing circle of a subtree neednot coincide with the root of the subtree,ﬂipping the drawing of a subtree(along the axis from the parent to the root of the subtree)might changeboth the aspect ratio and the angular resolution of the drawing.We showthat optimizing the angular resolution as well as the aspect ratio withrespect to this type of rooted ordered trees is reducible to the perfectmatching problem for bipartite graphs,which is solvable in polynomialtime.In addition,a related problem concerning the optimization of thedrawing area can be modelled as a speciﬁc type of nonlinear programmingfor which there exist several robust algorithms in practice.With a slightmodiﬁcation to the balloon drawing,we are able to generate the draw-ings of galaxy systems,H-trees,and sparse graphs,which are of practicalinterest.Article Type Communicated by Submitted RevisedRegular paper P.Eades and P.Healy January2006January2007*Corresponding author.Research supported in part by NSC Grant94-2213-E-002-086,and Research Grant95-EC-17-A-02-S1-049,Taiwan.C.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)432 1IntroductionGraph drawing addresses the issue of constructing geometric representations of graphs in a way to gain better understanding and insights into the graph struc-tures.If the given data is hierarchical(such as aﬁle system or a web graph), then it can often be expressed as a rooted tree.Among existing algorithms in the literature for drawing rooted trees,Reingold and Tilford[13]developed a popular method for drawing binary trees.The idea behind[13]is to recursively draw the left and right subtrees independently in a bottom-up manner,then shift the two drawings along the x-direction as close to each other as possible while centering the parent of the two subtrees one level up between their roots. Diﬀerent from the conventional‘triangular’tree drawing of[13],radial draw-ings[4]and balloon drawing[3,5,6,7,8]with respect to cone trees1[14]are also popular for visualizing hierarchical graphs.The interested reader is also referred to[9,10,11]for graph drawing techniques based on the idea of embed-ding trees/graphs in hyperbolic geometry to automatically obtain a deformation resulting in a focus+context navigation.Since the majority of algorithms for drawing rooted trees take linear time,rooted tree structures are suited to be used in an environment in which real-time interactions with users are frequent.A balloon drawing[3,6,7]of a rooted tree is a drawing having the following properties:(P1)all the children under the same parent are placed on the circumference of the circle centered at their parent,(P2)there exist no edge crossings in the drawing,and(P3)for the two edges on any path from the root node,the farther from the root an edge is,the shorter its drawing length becomes.Each subtree in the balloon drawing of a tree is enclosed entirely in a circle, which resides in a wedge whose end-point is the parent node of the subtree. Along any path from the root node,the radius of each circle reﬂects the number of descendents associated with the root node of the subtree.The ray from the parent to the root of the subtree divides the wedge into two sub-wedges. Depending on whether the two sub-wedge angles are required to be identical or not,a balloon drawing can further be divided into two types:drawings with even angles(see Figure1(a))and drawing with uneven angles(see Figure1 (b)).Another way to diﬀerentiate the two is that for the former(i.e.,the even angle case),it is required that the position of the root of a subtree coincide with the center of the enclosing circle of the subtree.The main aesthetic criteria on balloon drawing are angular resolution and aspect ratio.The angular resolution refers to the smallest angle between two 1Given a rooted tree,the cone tree method places the root of every subtree on the vertex of a cone and its children on the circumference of the base of the cone in3-D space such that there are no cones overlapping each other.A balloon drawing can be viewed as the projection of a cone tree on a plane.C.Lin and H.Yen,Balloon Drawings ,JGAA ,11(2)431–452(2007)433(a)Figure 1:Illustration of balloon drawings with (a)even angles and (b)uneven angles.adjacent edges incident to the common node in straight-line drawing,whereas the aspect ratio is deﬁned as the ratio of the largest angle to the smallest angle formed by two adjacent edges incident to the common node in straight-line drawing.A tree layout with a small aspect ratio often enjoys a very balanced view of a tree.It is not hard to observe that with respect to a rooted unordered tree,chang-ing the order in which the children of a node are listed aﬀects the angular reso-lution as well as the aspect ratio of the drawing.Hence an interesting question arises:How to ﬁnd an embedding of a rooted unordered tree such that the bal-loon drawing of the tree has the maximum angular resolution and the minimum aspect ratio?The ﬁrst part of this paper is devoted to answering the above question.We demonstrate an eﬃcient algorithm which is guaranteed to yield an optimal balloon drawing in terms of the (maximum)angular resolution and the (minimum)aspect ratio.Here we assume the balloon drawing to be of the even angle type.Now consider the case when the balloon drawing is of the uneven angle type.Allowing uneven angles introduces another dimension of ﬂexibility as far as optimizing the angular resolution and the aspect ratio is concerned.Even if the embedding (ordering)of a tree is given,ﬂipping the drawing of a subtree along the axis going through the parent and the root of the subtree might change the angular resolution as well as the aspect ratio of the drawing.Notice that in the uneven angle case,the angles on the two sides of the axis might not be equal.A related question is:How to ﬂip uneven angles in the balloon drawing of a rooted ordered tree to achieve optimality in angular resolution and aspect ratio?Notice that in the above,the embedding of the underlying tree is ﬁxed.As it turns out,we are able to reduce the above problem to that of perfect matching of bipartite graphs,which admits a polynomial time solution.In fact,the balloon drawing of the uneven angle case is derived from a process of minimizing the drawing area of balloon drawing of the even angle case subject to preserving properties (P1)-(P3)of balloon drawing (this will be illustrated in Section 4).There is,however,some work (e.g.,[5,8])suggestingC.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)434 that properties(P1)and(P3)may not be necessary for generating balloon drawings,i.e.,the nodes under the same parent need not be placed on the same circle circumference.As a result,we consider a problem related to optimizing the drawing area under a slightly modiﬁed version of balloon drawing,in which properties(P1)and(P3)of balloon drawing are relaxed by coloring the edges at the same level with the same color,i.e.,the level of an edge can be determined by observing its color[16].We are able to formulate the problem as a speciﬁc type of nonconvex nonlinear programming,for which there exist many robust algorithms in the literature for practical applications.In addition,minor extensions to the balloon drawing facilitate several in-teresting applications,including galaxy systems,H-trees,sparse graphs,among others.Our balloon drawing approach to galaxy systems gives an abstract lay-out of the related positions of bodies instead of their real positions and hence enjoys the advantage that the hierarchical structure of a galaxy system can be easily observed in such a layout.For more about visualizing the dynamic move-ments of bodies in a galaxy system due to their mutual forces,the reader is referred to[1]for the gravitational n-body problem.The rest of the paper is organized as follows.Some preliminaries are given in Section2.The algorithmic discussions for balloon drawings of even angles and uneven angles are given in Section3and Section4,respectively.Section5 discusses the related problem concerning the optimization of the drawing area. Section6gives some interesting applications.Finally,a conclusion is given in Section7.2PreliminariesA drawing of a graph G(=(V,E)where V is the set of nodes and E⊆V×V is the set of edges)on the plane is a mapping D from V to R2,where R is the set of real numbers.That is,each node v is placed at point D(v)on the plane, and each edge(u,v)is displayed as a line segment connecting D(u)and D(v).A rooted tree where the order of the subtrees is signiﬁcant andﬁxed is called an ordered tree;otherwise,it is called unordered.A star graph with n+1nodes is a rooted tree in which the root node is of degree n and the others are of degree one.Given a drawing of graph G,the angular resolution at node v refers to the smallest angle formed by two adjacent edges incident to v in the drawing of G. The angular resolution of a drawing of G is deﬁned as the minimum angular resolution among all nodes in G.The aspect ratio of a drawing of G is the ratio of the largest angle to the smallest angle in the drawing.The angular resolution (resp.,aspect ratio)of a drawing of G is in the range of(0◦,360◦)(resp.,[1,∞)). One of the objectives of this paper is,given a rooted unordered tree G,ﬁnding the optimum angular resolution and the optimum aspect ratio with respect to the so-called balloon drawing of G on the plane.There exist two models in the literature for generating balloon drawings of trees,as shown in Figure3(a)and(b),respectively.Given a node v,let r(v)C.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)435 be the radius of the drawing circle centered at v.If we require that r(v)=r(w)for two arbitrary nodes v and w that are of the same depth from the root of thetree,as shown in Figure3(a),then such a drawing is called a balloon drawingunder the fractal model[7].The fractal drawing of a tree structure means thatif l m and l m−1are the lengths of edges at depths m and m−1,respectively,then l m=γ×l m−1whereγis the predeﬁned ratio(0<γ<1)associated withthe drawing under the fractal model.Clearly,edges at the same depth have thesame length in a fractal drawing.Unlike the fractal model,the subtrees with nonuniform sizes(abbreviatedas SNS)model[3,6]allows subtrees associated with the same parent to residein circles of diﬀerent sizes,as shown in Figure3(b),and hence,the drawingbased on this model often results in a clearer display on large subtrees than thatunder the fractal model.Given a rooted ordered tree T with n nodes,a balloondrawing under the SNS model can be obtained in O(n)time in a bottom-upfashion(see[3,6])with the edge length and the angle between two adjacent edgesaccording to r=C/(2π)∼=(2 j R j)/(2π)andθj∼=(R j−1+free arc+R j)/r (see Figure2)respectively,where r is the radius of the inner circle centered atnode O,C is the circumference of the inner circle,R j is the radius of the outercircle enclosing all subtrees of the j-th child of O,and R O is the radius of theouter circle enclosing all subtrees of O.Since there exists a gap between C andthe sum of all diameters,we can distribute to everyθj the gap between themevenly,denoted by free arc.(a)(b)Figure2:The SNS model.(a)Node O is not the root,and the edge between O and its parent goes through a circle with radius R min;(b)is a star graph centered at c0.Assume that node O has n children.When computing r andθj in(a),one should notice that R min is viewed as R n+1,and hence,(a)can be viewed as(b)where node O has n+1children.In general,it suﬃces to consider the case of(b)when computing r andθj.Note that the trees considered in[3,6]are ordered.Since the fractal model assumes that all the angles between neighboring edges emanating from a com-C.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)436 mon node are the same,changing the ordering of the subtrees of a node at any level does not aﬀect the angular resolution(nor the aspect ratio)of the drawing. Under the SNS model,however,the ordering of subtrees is critical as far as the angular resolution and the aspect ratio are concerned.Our goal is to devise an algorithm forﬁnding the balloon drawing of a rooted unordered tree(under the SNS model)for which the angular resolution and the aspect ratio are optimized.3Finding the balloon drawing with optimum angular resolution and optimum aspect ratio First,we consider the following way of drawing a star graph with circles of nonuniform size attached to the children of the root(see Figure2(b)for an example).Deﬁnition1The balloon drawing of a star graph with children of nonuniform size is a drawing in which1.circles associated with diﬀerent children of the root do not overlap,and2.all the children of the root are placed on the circumference of a circlecentered at the root.Let S be a star graph with n+1nodes{c0,c1,...,c n},where c0is the root. It can easily be seen from Figure2(b)that,in a balloon drawing of S,the circle centered at the root is divided into n wedges(or slices)each of which accommodates a circle associated with a child of c0.Let w i(resp.,w′i)be the angle between rays−−→l i−1and−−→c0c i(resp.,−→l i and−−→c0c i).The balloon drawing is said to be of even angle if w i=w′i,for all1≤i≤n.That is,−−→c0c i divides the respective wedge into two equal sub-wedges;otherwise(i.e.,the requirement w i=w′i is dropped),the drawing is said to be of uneven angle.In this section, we only consider balloon drawings of even angle.More will be said about the uneven angle case in Section4.Letθi,1≤i≤n,be the size of the wedge angle enclosing the circle centered at node c i.(In Figure2(b)θi=w i+w′i=2w i,assuming the even angle case.) An ordering of the children of c0is simply a permutationσof{1,...,n},which speciﬁes the placements of nodes c1,...,c n(and their associated circles)along the circumference of the circle centered at c0in the balloon drawing.Moreprecisely,the children are drawn in the order of cσ1,cσ2,...,cσn,in which cσiand cσi⊕1,1≤i≤n,are neighboring nodes.2With respect toσ,the size ofthe angle between−−→c0c i and−−−−→c0c i⊕1isθσi+θσi⊕12.Hence,the angular resolution(denoted by AngReslσ)and the aspect ratio(denoted by AspRatioσ)areAngReslσ=min1≤i≤n{θσi+θσi⊕12}2i⊕1denotes(i mod n)+1C.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)437 andAspRatioσ={max1≤i≤n{θσi+θσi⊕12} min1≤i≤n{θσi+θσi⊕12}}LetΣbe the set of all permutations of{1,...,n}.In what follows,we shall design an eﬃcient algorithm toﬁnd a permutation that returnsoptAngResl=maxσ∈Σ{AngReslσ}andoptAspRatio=minσ∈Σ{AspRatioσ}The optAngResl is said to involve anglesθσi andθσi⊕1if i is the value thatminimizes AngReslσwith respect to the optimal permutationσ.It should be noticed that the permutation witnessing optAngResl or optAspRatio may not be unique.For notational convenience,we order the set of wedge anglesθ1,...,θn in ascending order as eitherm1,m2,...,m k−1,m k,M k,M k−1,...,M2,M1(1) if n is even,orm1,m2,...,m k−1,m k,mid,M k,M k−1,...,M2,M1(2) if n is odd,for some k where m i(resp.M i)is the i-th minimum(resp.maxi-mum)among all,and mid is the unique median.We deﬁneαij=M i+m j2,1≤i,j≤k.Recall from Figure2(a)that the drawing of the subtree rooted at a node O is enclosed in a circle centered at O.By abstracting out the details of each of the subtrees associated with the children of O,the balloon drawing of the subtree at O can always be viewed as a balloon drawing of a star graph with children of nonuniform size rooted at O,regardless of the depth at which O resides.In addition,even if we alter the ordering of the children of O,the size of the outer circle bounding all the children of O remains the same;hence,the optimization of each of the subtrees at depth k does not aﬀect the optimization of their parent at depth k−1.In view of the above,optimizing the angular resolution as well as the aspect ratio of a balloon drawing of a rooted unordered tree can be carried out in a bottom-up fashion.As a result,it suﬃces to investigate how to optimize the angular resolution and the aspect ratio of balloon drawing with respect to star graphs.We give the drawing algorithm in Procedure1that achieves optimal angular resolution and aspect ratio under the SNS balloon drawing model.Theorem1Procedure1achieves optimality in angular resolution as well as in aspect ratio for star graphs.C.Lin and H.Yen,Balloon Drawings ,JGAA ,11(2)431–452(2007)438Procedure 1OptBalloonDrawingInput:a star graph S with n child nodes of nonuniform size.Output:a balloon drawing of S optimizing angular resolution and aspect ratio.–Sort the set of the wedge angles (accommodating the n nonuniform circles)into ascending order as mentioned in Equations (1)and (2).–Depending on whether n =2k or 2k +1and whether k is odd or even,output (M 1,m 2,M 3,m 4,...(,mid ),...,M 4,m 3,M 2,m 1)with the corresponding drawing illustrated below:(Note that M 1and m 1are adjacent.)(a) If n = 2kand k is odd.(b) If n = 2k and k is even.(d) If n = 2k + 1 and k is even.(c) If n = 2k + 1 and k is odd.3...m 3......11m 3113...Proof:In what follows,we only consider the caseσ=(M 1,m 2,M 3,m 4,...,M k −1,m k ,mid,M k ,m k −1,...,M 4,m 3,M 2,m 1)i.e.,n =2k +1and k is odd,and assuming that the sizes of angles are all distinct (break ties arbitrarily);the remaining cases are similar (in fact,simpler).Recall that αij =M i+m j 2.One can easily see the following properties of σ:Property (1).for each i ∈{2,...,k },the angles of sizes (mid +m k )/2,α(i −1)i =(M i −1+m i )/2,(M k +mid )/2,and αi (i −1)=(M i +m i −1)/2are included in σ;Property (2).the size of the smallest angle of σmust be (mid +m k )/2orαj (j −1),while the size of the biggest angle of σmust be (M k +mid )/2or α(l −1)l for some j,l ∈{2,...,k }.The reason behind Property (2)is that all the angles consecutively appearing in σhave the following relationship:α12>α32<α34>...>αj (j −1)<...<α(l −1)l >...>αk (k −1)<M k +mid 2>mid +m k 2<α(k −1)k >...>α43<α23>α21<M 1+m 12<α12where,for each j,l ∈{2,...,k },αj (j −1)and (mid +m k )/2are wrapped between ‘>’and ‘<’,while α(l −1)l and (M k +mid )/2are wrapped between ‘<’and ‘>’.C.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)439Supposeδis the permutation that witnesses optAngResl.From Property (2),the minimum angular resolution ofσmust be eitherαi,i−1,for some2≤i≤k,or(mid+m k)/2.In what follows,we only consider the case when the minimum angular resolution ofσisαi,i−1=(M i+m i−1)/2;the other case can be proved similarly.Now if M i is a neighbor of m i−1inδ(i.e.,the optimal permutation),then δandσhave the same angular resolution and optAngResl=αi,i−1.Because, otherwise,•if optAngResl<αi,i−1(which is the angular resolution ofσ),then this contradicts thatδis the optimal.•if optAngResl>αi,i−1,then this is impossible becauseδhas an angle of sizeαi,i−1=(M i+m i−1)/2.Hence,σis optimal as well.On the other hand,suppose x and y(x<y)are the two neighbors of m i−1in δand neither one is M i,then both x and y must be greater than M i;otherwise, the angular resolution ofδis smaller than(M i+m i−1)/2–contradictingδbeing optimal.Also note that optAngResl≥(m i−1+x)/2.Now if we look at a partition of the set of wedge angles of S as follows:m1<...<m i−1<... <M i<...<x<...<y<...M1.R A R B R CR A contains i−2elements,which must be connected to at least i−1elements of R C;otherwise,the angular resolution inδbecomes less than(M i+m i−1)/2–a contradiction.R C originally contains i−1elements.However,x and y are the two neighbors of m i−1,meaning that together with m i−1they are tied together and cannot be separated.So eﬀectively only‘i−2’elements of R C can ﬁll the i−1neighbors of R A,which is not possible.We have a contradiction. What the above shows is that for a permutationδto be optimal,a neighbor of m i−1is M i.Hence,the angular resolution ofδis≤(M i+m i−1)/2.Since δwitnesses optAngResl,optAngResl=(M i+m i−1)/2,meaning thatσalso produces optAngResl.The above implies that optAngResl must be either(mid+m k)/2orαi(i−1) for some i∈{2,...,k},which is always included in the circular permutationσproduced by Procedure1.Similarly,we can prove that the minimum size of the largest angle of any drawing must be(M k+mid)/2orα(j−1)j for some j∈{2,...,k},which is also always inσ.Sinceσsimultaneously possesses both the maximum size of the smallest angle and the minimum size of the largest angle of any drawing,σalso witnesses the optimum aspect ratio.2Using Procedure1,the drawing of a rooted unordered tree which achieves optimality in angular resolution and aspect ratio can be constructed eﬃciently in a bottom-up fashion.C.Lin and H.Yen,Balloon Drawings ,JGAA ,11(2)431–452(2007)440(a) using fractal model.(b) using SNS model, ass- -uming an ordered tree.(c) using our method, assu--ming an unordered tree.A B CD E F GA B C DE F G A B C D EF G |V | max degree depth fractal SNS our fractal SNS our1000 20 4 18 3.50 6.33 1 13.496 4.672tree information angular resolution aspect ratioo o o Figure 3:Three drawings of a tree and their statistics.Finally,in order to give a sense perception on the advantages of our algo-rithm,the algorithm was implemented with a simple experimental result shown in Figure 3.The drawing in Figure 3(a)is based on the fractal model,which displays a drawing in which the sizes of angles spanned by adjacent edges are identical,and hence,Figure 3(a)has the best angular resolution and aspect ratio.The major drawback of fractal drawing,as can be seen in Figure 3(a),is that visibility deteriorates considerably as we move towards deeper subtrees of complicated structures.In order to observe more complicated subtrees clearer and easier,one may apply the SNS model (with respect to an ordered tree)to yield Figure 3(b).(Note that Figure 3(b)has the same ordering as that in Fig-ure 3(a).)To a certain extent,the SNS model sacriﬁces the angular resolution and the aspect ratio in order to gain better visibility for displaying complicated subtree structures.If the ordering of subtrees is allowed to be altered,our optimization algorithm has the ability to optimize the angular resolution and the aspect ratio under the SNS model on the balloon drawing of the rooted unordered tree,as shown in Figure 3(c).As a direction for future research,it is of interest to compare (experimen-tally)our work presented in this section with the technique of bubble tree draw-ing ,which,according to [5],also exhibits good results in angular resolution.(Note,however,that [5]only deals with ordered trees,and their goal to ﬁnd a good drawing on the tradeoﬀbetween the angular resolution and the edge lengths is diﬀerent from our work presented in this section.)It is also worthy of pointing out that the algorithm presented in [5]diﬀers from the SNS model in the sense that they alow edges to have bends in order to optimize the use of screen space.In this regard,bubble tree drawing is relevant to our discussionC.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)441 in Section5(dealing with optimizing the drawing area)and certainly deserves to be compared with our technique in greater detail in the future.4Balloon drawing with uneven anglesThe area of a balloon drawing can be measured by the size of the circle en-closing the balloon drawing.Minimizing the area of a drawing is an important issue because any drawing needs to be rendered on a limited region.A careful examination of the approach investigated in Section3suggests that the area of balloon drawing generated by the SNS model may not be minimal.Part of the reason is the involvement of the so-called free arc described in the SNS model and Figure2(b),serving for the purpose of separating the enclosing circles of two neighboring subtrees.A more subtle point regarding the‘waste’of drawing space is illustrated in Figure4,in which Figure4(a)shows the drawing of a tree under the SNS model.Let T v be the subtree rooted at v.Based on the approach discussed in Section3,T v resides in a circle centered at v and the circle is included in a wedge in which the ray from O to v cuts the wedge into two sub-wedges of identical size(i.e.,ω1=ω2).By limiting the drawing to the area formed by two lines(see t1and t2in Figure4(a))tangent to the drawing of T v,the drawing area is reduced.That is,the new wedge(in which the drawing of T v resides)is now spanned by lines t1and t2with the size of the wedge angle equalsω3+ω4.Furthermore,the ray−→Ov cuts this new wedge into two possibly uneven parts(i.e.,ω3need not be equal toω4).(a) Original drawing(b)(c)Figure4:An approach for minimizing the drawing area where each shaded circle is a star graph.Allowing uneven angles tends to release extra space between the drawings of neighboring subtrees,in comparison with the case discussed in Section3when only even angles are permitted.The presence of such extra space allows us to move the position of each subtree inwards(i.e.,towards the root node O)which, in turn,reduces the drawing area as Figure4(b)shows.The drawing area canC.Lin and H.Yen,Balloon Drawings,JGAA,11(2)431–452(2007)442 further be reduced by shrinking the free arc on the bottom of Figure4(b).Theﬁnal drawing is shown in Figure4(c)which obviously has a smaller drawingarea(compared with Figure4(a)).However,the angular resolution and theaspect ratio of the drawing might deteriorate as Figure4(c)indicates.With respect to Figure4(c),it is interesting to observe that the angularresolution and the aspect ratio might change if weﬂip the drawing of subtreeT v along the axis−→Ov(i.e.,swappingω5andω6).In view of the above,a natural question is to determine how the two possibly uneven angles associated with asubtree are arranged in order to achieve an optimal aspect ratio and angularresolution,assuming the ordering of subtrees in the drawing isﬁxed.Such aquestion can be formulated as follows.Suppose O is the root of a star graph with n subtrees rooted at c1,...,c nwhich are listed in a counterclockwise fashion.Suppose the sizes of the twoangles associated with subtree c i(1≤i≤n)are w0i and w1i.See Figure5for an example.Then the sequence of angle sizes encountered along the circlecentered at O can be expressed as:{w t11,w t′11,w t22,w t′22,...,w t i i,w t′ii,...,w t n n,w t′nn}c1c2...c i...c n,(3)where t i,t′i∈{0,1}and t i+t′i=1.With respect to the above,the angular resolution isAngResl=min1≤i≤n{w t′i i+w t i⊕1i⊕1}and the aspect ratio isAspRatio=max1≤i≤n{w t′i i+w t i⊕1i⊕1} min1≤i≤n{w t′i i+w t i⊕1i⊕1}.The problem then boils down to assigning0and1to t i and t′i(1≤i≤n)inorder to optimize AngResl and AspRatio.Note that the two values(either(0,1)or(1,0))of(t i,t′i)correspond to the two conﬁgurations of the drawing of thesubtree associated with c i and one is obtained from the other byﬂipping alongthe axis−−→Oc i.To give the reader a better understanding of how the angular resolution is aﬀected in the uneven angle case,consider Figure5.In Figure5(a),the size of the angle between subtrees c1and c2is w11+w02.Now if weﬂipthe drawing of subtree c2as shown in Figure5(b),the angle between subtreesc1and c2becomes w11+w12.Consider the following problems:The Aspect Ratio(resp.,Angular Resolution)Problem:Given theinitial drawing of a star graph(with uneven angles)speciﬁed by Equation(3)and a real number r,determining the assignments(0or1)for t i andt′i(1≤i≤n)so that AspRatio≤r(resp.,AngResl≥r);return false ifno such assignments exist.。