I have known many people who feel that a tree should be left to its own way, neither trimmed nor pruned, and certainly never taken out! I may have felt somewhat those passions myself when my interest in plants was a budding love. And still, when trees are cut simply to clear a spot for nothing, or to build some poor excuse for architecture, I am always saddened. You can assume there is a but. But, trees, like all plants have a lifespan. They have disease which makes them dangerous to surrounding plants and structures. And they have bad habits. They rarely drink too much, nor do they often smoke (except at the hand of careless human friends) - but they, like children will often develop a little habit here and there that doesn't seem like much - till one year the neighbor's prized roses have been crushed by an errant branch, or there is a pile of rotting fruit in their driveway.
Some time ago, I wrote about
pruning shrubs, perennials and flowers. The trimming of trees follows similar rules, but with trees, the pruning will create a permanent shape. Perennials that are just whacked down in spring or fall, will grow as they want, but the shape of a tree and its ability to do its best work - flowering, bearing fruit, providing shade, being beautiful - is dependent on the work of the gardener. There is something so artistic and satisfying about pruning a young tree (because while old trees need pruning too, that's more a hacksaw\chainsaw event, than a pair of hand clippers and small saw like a younger one). Leaves, flowers and fruit need air and light, so when the middle of a tree has cross branches - branches that grow into the tree mass rather than outward-the tree looks tangled and messy when there are no leaves on it. A tree with an aesthetically pleasing shape when it's bare, will also be prettier and stronger when it's beautifully leafed out. (Is this like people? Because I look soooo much better with leaves, lots of leaves.)

This drawing shows some reasons trees are pruned. In snowy climates prune broken branches so the snow won't do it for you. If a branch looks dead - cut it off! If one or more are growing crossways, cut them right at the trunk, in the spring while you can see clearly - and keep stepping back to see if all the branches are growing out.
Some shooters growing straight up from a branch - off with their heads! A pretty big one growing from the trunk, but into the tree or toward the ground - off! It's so satisfying! So any branch that is unsafe, sick or just looks bad, should be your springtime enemy.
Trees certainly do not need to have some arbitrary shape to be beautiful. But healthy is nearly always prettier than unhealthy. And each tree species - especially newer cultivars - has it's own ideal shape. Like this flowering pear, which was bred to have a flame pattern.
Sometimes one that grows unusually can be beautiful anyway, especially if it has room to be seen and grow without interfering, or interference from, other trees or structures. Like this one!
The touchiest of all tree care topics is when a tree should come out - or never be planted in the first place. Many neighborhood feuds and community ire has been ignited over trees trimmed, taken out or not planted.
My version of this eternal dispute is that if a tree is fast growing (i.e.weak structure - which is virtually synonymous), plant at your own own risk and know that one day it may well have to be taken out, or radically trimmed. Poplars, willows, elms, Russian Olive and others that grow fast are notoriously weak wood and susceptible to disease and bugs. A weeping willow, near water and from a distance is a beautiful thing.

But in a normal yard with little ground water nearby, it will overpower homes, power lines and other plants, and drops branches all over the yard with the slightest breeze. Chinese Elm is very sickly and is almost always host to little worms that land in the hair of innocent children causing trouble for their mothers. And they are an official weed in most places. I love the dark bark and beautiful silvery green of Russian Olive along a riverbank. While I'm driving by. Fast. But they are a menace to water managers and people with allergies, and they too are weak and likely to break.
Many cultivars of willow are currently popular with yard builders and homeowners. The Star Willow is a fine example. It's hard to even find in research because it is willow grafted to a short, sturdy trunk, which grows in a very round shape, and in spring the branch tips are white so it's very pretty. Until year 2,when it is completely out of control and the original shape is gone unless you are a vigilant pruner. Or the coral bark willow, which loses its red bark if not completely cut down each year. Still it grows to 8 feet around and 20 feet high - so, great for the North 40 where it will define your acreage. Another new cultivar which grows fast and tall, has some of its parent weakness bred out - Swedish Aspen. It is a poplar/aspen hybrid and it still grows fast, but resists much of the disease of both parents. Not a pretty tree, but very good for privacy and hiding the neighbors' garage.
Sadly, many trees just have a lifespan, and they need to be replaced. Sycamores and Maples do not get what they need from our Intermountain West conditions. They may live for years, but if you bought your home at the end of the 40 or 50 or even 80 year life, it's hard to say, time to go old friend! Topping (boo) and cutting around power lines may work for a while, but at some point it's just distressing to see a mangled tree stick around for a scrap of shade. Sometimes a tree is just too old or too sick or too weak, and it has to go. There will undoubtedly be, in each such case, someone who wants to hang on to it anyway.
Great trees for ornamental and shade use in yards for another day. Here is one of my favorites, the Honey Locust. Tiny leaves that seem to just disappear, filtered shade, and a lovely shape.