Can you plant trees by planting its branches?

yes a willow tree 

Technically this is what you do, but it helps to use a "rooting agent". Even so, there are many failures in trying to grow plants from CLIPPINGS. There are also certain plant species that are harder to grow in this method while others are more adapt to growing stringers from a life plant. To prevent growing a "bush" you need to prude the plant as it grows.

There are ways to do it.  But you may not end up with a tree. But a bush.  I use to have several nice cherry bush's.  That were the limbs from a cherry tree. That were rooted then planted. 

How do you root a tree branch: 

Pinch off the soft tip of the stem, then dip the cut end into a pile of rooting hormone powder. Poke a hole into a pot of compost to make room for the stem, and plant it so that the first set of leaves is right above the surface of the compost. Water the pot and place it in a plastic bag.

You should give us the kind of tree you want to plant by its branches. We will give you the exact answer.

Some kinds of trees can planted with its branches. Others aren't.

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certain species yes. i don't grow tree's but what we typically use for other plants with this type of method is clonex which is a rooting agent. i believe it tricks the plant into producing roots faster in the cuttings. Clonex is a product but there are other brands you can use as well. 

we do this in the cannabis industry a lot saves time and money and to take clones from certain plants we want to save genetics for instead of making mother plants 

If it works at all, it would be on SMALL branches/twigs.

There is also "Layering" as a method of propagation. 

Some will do that, willow particularly is known for that ability.  Others won't. 

I've done that myself, just heeled some fresh young offcuts from a willow into the ground and now we have a fine stand of trees.  I've heard of it being done with hazel too, though I've never seen that personally.

Yes, there are some plants like hibiscus which grow in this way. But not what all of them

No, it will not grow.