is not a growth node but is an ancestor of a growth node. A *dead node* is a node of $T$ that is neither a living nor a growth node, that is, it has no descendant that is a growth node. Every pair of nodes in this tree has a *nearest common ancestor* (NCA). When we have to decide what parent to use for a new node in the tree, we look at the maximum value on the path in the tree between the parents and their NCA (excluding the value of the latter). A *face* of the tree is the area enclosed by the segment between two horizontally or vertically neighboring growth nodes (without one being the parent of another) and the paths to their NCA. The unique *sink* of a face is the node of the grid that is in the lowest column and row of all nodes on the face. Fig. \[fig:sinksandshortcuts\] (a-b) shows some examples of faces and their sinks.

![(a) Face of tree (gray area) with its unique sink (solid dot). A dashed line represents a dead path. (b) Two adjacent faces with some shortcuts indicated. (c) Tree with 3 faces. Solid dots indicate growth 