Let $α \in [0, 1)$ be a constant. For a rooted binary tree $T$ and a node $x$ in $T$, we denote by $|x|$ the number of nodes in the subtree of $T$ rooted at $x$ (if $x$ = $NIL$ then $|x|$ = $0$). We say that $T$ is $α$-good if, for every node $x$ in the tree with children $y$ and $z$, it holds that $|y|$ − $|z|$ ≤ $α|x|$. Show that an $α$-good tree with n nodes has height $O(\log n)$.
I tried using induction for this but made no progress.
Some other observations I made:
$|x| = |y| + |z| + 1$
$\le 2|z| - (|z| - |y|) + 1 $ (WLOG assume $|y|\le|z|$)
$\le 2|z| - ||z| - |y||+1$
$\le 2|z| - \alpha |x| + 1$
$ = 2|z| - \alpha |y| - \alpha |z| + (1 - \alpha)$
I'm not sure if it helps.