We revisit weight-balanced trees, also known as trees of bounded balance. Invented by Nievergelt and Reingold in 1972, these trees are obtained by assigning a weight to each node and requesting that the weight of each node should be quite larger than the weights of its children, the precise meaning of ``quite larger'' depending on a real-valued parameter $\gamma$. Blum and Mehlhorn then showed how to maintain them in a recursive (bottom-up) fashion when $2/11 \leqslant \gamma \leqslant 1-1/\sqrt{2}$, their algorithm requiring only an amortised constant number of tree rebalancing operations per update (insertion or deletion). Later, in 1993, Lai and Wood proposed a top-down procedure for updating these trees when $2/11 \leqslant \gamma \leqslant 1/4$.

Our contribution is two-fold. First, we strengthen the requirements of Nievergelt and Reingold, by also requesting that each node should have a substantially larger weight than its grandchildren, thereby obtaining what we call grandchildren-balanced trees. Grandchildren-balanced trees are not harder to maintain than weight-balanced trees, but enjoy a smaller node depth, both in the worst case (with a 6 \% decrease) and on average (with a 1.6 \% decrease). In particular, unlike standard weight-balanced trees, all grandchildren-balanced trees with $n$ nodes are of height less than $2 \log_2(n)$.

Second, we adapt the algorithm of Lai and Wood to all weight-balanced trees, i.e., to all parameter values $\gamma$ such that $2/11 \leqslant \gamma \leqslant 1-1/\sqrt{2}$. More precisely, we adapt it to all grandchildren-balanced trees for which $1/4 < \gamma \leqslant 1 - 1/\sqrt{2}$. Finally, we show that, except in limit cases (where, for instance, $\gamma = 1 - 1/\sqrt{2}$), all these algorithms result in making a constant amortised number of tree rebalancing operations per tree update.