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The phase transition in the number partitioning problem (NPP), i.e., thetransition from a region in the space of control parameters in which almost allinstances have many solutions to a region in which almost all instances have nosolution, is investigated by examining the energy landscape of this classicoptimization problem. This is achieved by coding the information about theminimum energy paths connecting pairs of minima into a tree structure, termed abarrier tree, the leaves and internal nodes of which represent, respectively,the minima and the lowest energy saddles connecting those minima. Here we applyseveral measures of shape (balance and symmetry) as well as of branch lengths(barrier heights) to the barrier trees that result from the landscape of theNPP, aiming at identifying traces of the easy/hard transition. We find that itis not possible to tell the easy regime from the hard one by visual inspectionof the trees or by measuring the barrier heights. Only the {\it difficulty}measure, given by the maximum value of the ratio between the barrier height andthe energy surplus of local minima, succeeded in detecting traces of the phasetransition in the tree. In adddition, we show that the barrier trees associatedwith the NPP are very similar to random trees, contrasting dramatically withtrees associated with the $p$ spin-glass and random energy models. We alsoexamine critically a recent conjecture on the equivalence between the NPP and atruncated random energy model

Phase transition and landscape statistics of the number partitioning problem