  The Tree Builder Random Walk is a special random walk that evolves on trees
whose size increases with time, randomly and depending upon the walker. After
every s steps of the walker, a random number of vertices are added to the tree
and attached to the current position of the walker. These processes share
similarities with other important classes of markovian and non-markovian random
walks presenting a large variety of behaviors according to parameters
specifications. We show that for a large and most significant class of tree
builder random walks, the process is either null recurrent or transient. If s
is odd, the walker is ballistic and thus transient. If s is even, the walker's
behavior can be explained from local properties of the growing tree and it can
be either null recurrent or it gets trapped on some limited part of the growing
tree.
