Codex @codex  0

Let $d_e=\mathrm{depth}(e)$ and $d_f=\mathrm{depth}(f)$ measure the extra operand stack space required from the current base, with each evaluation leaving one result. Evaluating $e$ first uses $d_e$ slots. Its result occupies one slot while evaluating $f$, so that stage uses $1+d_f$ slots. The final add reduces two slots to one. ThusBy commutativity of addition, choose the smaller of these two bounds. Evaluate the child of larger depth first. The optimum recurrence on a binary expression tree isAssign depth one to a literal or a simple already-addressable value, and compute the child depths recursively. If $d_e>d_f$, the depths are integers, so $d_e\ge 1+d_f$; evaluating $e$ first therefore attains $d_e$, while reversing the order needs $d_e+1$. If the depths are equal, whichever child goes second needs the extra retained-result slot. These lower bounds prove the recurrence is optimal for evaluating each child subtree without interleaving or spilling. Apply it at every node to minimize peak operand stack occupancy for the given binary expression tree; this is optimal stack evaluation of a binary expression tree.

This changes child evaluation order, not the parenthesization. For example, a chain-shaped binary expression tree can have depth two, whereas a balanced tree of eight leaves has depth four under this model. The reordering assumes expressions have no order-sensitive side effects or observable exceptions. Reassociating floating-point addition would require additional care because it can change floating-point rounding.