How to determine if binary tree is balanced?

Stumbled across this old question while searching for something else. I notice that you never did get a complete answer.

The way to solve this problem is to start by writing a specification for the function you are trying to write.

Specification: A well-formed binary tree is said to be “height-balanced” if (1) it is empty, or (2) its left and right children are height-balanced and the height of the left tree is within 1 of the height of the right tree.

Now that you have the specification, the code is trivial to write. Just follow the specification:

    return (tree is empty) or 
           (IsHeightBalanced(tree.left) and
            IsHeightBalanced(tree.right) and
            abs(Height(tree.left) - Height(tree.right)) <= 1)

Translating that into the programming language of your choice should be trivial.

Bonus exercise: this naive code sketch traverses the tree far too many times when computing the heights. Can you make it more efficient?

Super bonus exercise: suppose the tree is massively unbalanced. Like, a million nodes deep on one side and three deep on the other. Is there a scenario in which this algorithm blows the stack? Can you fix the implementation so that it never blows the stack, even when given a massively unbalanced tree?

UPDATE: Donal Fellows points out in his answer that there are different definitions of ‘balanced’ that one could choose. For example, one could take a stricter definition of “height balanced”, and require that the path length to the nearest empty child is within one of the path to the farthest empty child. My definition is less strict than that, and therefore admits more trees.

One can also be less strict than my definition; one could say that a balanced tree is one in which the maximum path length to an empty tree on each branch differs by no more than two, or three, or some other constant. Or that the maximum path length is some fraction of the minimum path length, like a half or a quarter.

It really doesn’t matter usually. The point of any tree-balancing algorithm is to ensure that you do not wind up in the situation where you have a million nodes on one side and three on the other. Donal’s definition is fine in theory, but in practice it is a pain coming up with a tree-balancing algorithm that meets that level of strictness. The performance savings usually does not justify the implementation cost. You spend a lot of time doing unnecessary tree rearrangements in order to attain a level of balance that in practice makes little difference. Who cares if sometimes it takes forty branches to get to the farthest leaf in a million-node imperfectly-balanced tree when it could in theory take only twenty in a perfectly balanced tree? The point is that it doesn’t ever take a million. Getting from a worst case of a million down to a worst case of forty is usually good enough; you don’t have to go all the way to the optimal case.

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