y minimizes Σ dᵤ φ(yᵤ), with total mass k.
When xℓ − yℓ ≥ 3/2, one miner moves through the nearest suitable ancestor to an underloaded leaf.
Extend with at least 2 miners. Fork into 2 to xℓ − 1 children. Zero descendants deletes a leaf.
Vertical height represents edge length. Nodes at the same distance from the root line up horizontally. Curved links separate the branches; faint grey labels show edge lengths.
From Polylogarithmic Collective Tree Exploration
R. Cosson & L. Massoulié