In a taller tree, insert descends to a leaf, and if that leaf splits the separator has to climb back up to its parent. Today you make insert recursive so a leaf split is absorbed by a parent that still has room.
Insert into the correct leaf of a multi-level tree, and when it splits, add the separator to its parent.
Insert becomes recursive: descend into the child that should hold the key, and let that call report back whether it split. If a child leaf splits, it returns its separator and the id of its new right half, and the parent splices them in - inserting the separator among its keys and the new child among its children, at the position the routing already picked. The key stays sorted, the child sits just to the right of its separator.
Today the parent always has room, so it absorbs the new separator without splitting itself - one idea at a time. That keeps the focus on the plumbing: how a split result travels up one level and gets stitched into the parent. What happens when the parent is also full - so the split cascades further up, and eventually grows a new root - is the next lesson, and it reuses the internal split you already wrote.
// insert returns (sep, newRightID, didSplit) to the caller above.func (t *Tree) insert(id, key, val uint64) (uint64, PageID, bool) {// leaf: insert; if overflow, split and return its separator up// internal: recurse into childIndex; if the child split, splice// the returned separator+child into this node's Keys/Children// (this node has room today, so it does not split yet)}