Maximum height of a quad tree

Question

If we have a quad tree where each node must have 0 or 4 children, is there an expression that can me the maximum height of a quad tree with $n$ nodes?

Answer 1

A tree of height $m$ where at each level only one node has children will have $1+4(m-1)$ nodes, assuming a tree with only one node has height one. The height will be $m=\frac{n+3}{4}$ for $n$ nodes. If your number of nodes is not $1 \pmod 4$, it won't work as each branch adds 4 nodes.