The basic idea is to walk along the outside of the tree hierarchy and to number the left and right side of every node along the way:

While this is rather unintuitive it allows to:

• Get the sub tree of an node by selecting all nodes whose left value is between the left and the right value of the top node of the subtree
• Get the path of a node by selecting all nodes whose left value is smaller than the left value of the node in question and whose right value is larger than its right value
• Determine the number of children of a node by calculating:
  ( right value – left value -1 ) / 2

The disadvantage of this approach is that the addition of a single node leads to changes in all nodes to the “right” of it. After pondering the problem for a while I came up with a variant of the algorithm that gets rid of that. Instead of using simple consecutive numbers for the left and right values I wrote an algorithm which spreads all the left/right values over a fixed range of numbers. The top node starts with a fixed left and right value and children are inserted by splitting that interval into smaller sub-intervals. This makes it possible to not change the key values on inserting and do that insert with a single writing operation.

Links:

1. Postgresql stored procedure implementing the keyspace variant
2. Modified preorder tree traversal algorithm
   Another explanation of the modified tree traversal algorithm with code samples in PHP