Height and balance are important things to think about when using tree data structures. This is especially true for types like binary trees, binary search trees (or BSTs), AVL trees, and red-black trees.

First, height is key to how well we can do tasks like searching, adding, and removing items from these trees. For example, in a simple binary tree, the height can get pretty big, up to $n$ in the worst situation. This means that operations can take a long time, running in $O(n)$ time, which is not great. But in a binary search tree, if it's balanced, the height is usually around $\log(n)$. This balance helps operations run much faster, in $O(\log n)$ time.

Second, balance is just as important. It is a major feature of AVL trees and red-black trees. An AVL tree keeps a strict balance, meaning that the heights of its two child trees can only differ by one. This keeps the height low, which leads to faster operations at $O(\log n)$ for adding, removing, and searching for items. Red-black trees are a bit less strict about how they balance, but they still keep the height low, so they can work just as efficiently.

In short, how height and balance work together is really important for how well trees perform. Trees that keep their heights low through balance can work much better. On the other hand, if a tree is unbalanced, it can slow down performance and make handling data difficult. So, it’s really important to choose the right type of tree based on what you need to do for the best results in data structures.

Footnote:
[1] In this case, $n$ means the number of nodes in the tree.