Proof for height of a binary tree

I keep forgetting the standard result of the height of a binary tree. I figured, writing out the proof on my own might help me remember the result.

We assume that the tree is a full binary tree for simplicity. A full binary tree is defined as follows[1]:

A binary tree in which each node has exactly zero or two children.

A binary tree starts with the root node at level one(we consider levels to be 1-indexed).
It’s two children form the next level.
For any given level \(i\)(where \(i > 0\)), the number of nodes at the level \(n_i\) can be defined as:

For any given height \(h\), the total number of nodes in the tree \(n\) is given by:

The number of nodes in a tree is given by a geometric progression with common ratio \(2\) and \(h\) number of terms.
\(n = 2^h - 1\)
Thus, \(h = \log_2 {(n + 1)}\). \(\blacksquare\)

References

1. https://xlinux.nist.gov/dads/HTML/fullBinaryTree.html