Proof with try tauto.
intros.
split;repeat intro. icase n.
destruct (eq_dec sL bot)... subst.
destruct (eq_dec sR bot)... subst.
apply tree_avg_zero in H...
apply tree_avg_zero in H...
Qed.

Lemma tree_avg_bound: forall sL sR s n,
avg n sL sR = Some s -> (height s <= n)%nat.