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The Tree Builder Random Walk is a special random walk that evolves on treeswhose size increases with time, randomly and depending upon the walker. Afterevery s steps of the walker, a random number of vertices are added to the treeand attached to the current position of the walker. These processes sharesimilarities with other important classes of markovian and non-markovian randomwalks presenting a large variety of behaviors according to parametersspecifications. We show that for a large and most significant class of treebuilder random walks, the process is either null recurrent or transient. If sis odd, the walker is ballistic and thus transient. If s is even, the walker'sbehavior can be explained from local properties of the growing tree and it canbe either null recurrent or it gets trapped on some limited part of the growingtree.

Tree Builder Random Walk: recurrence, transience and ballisticity