Automatic organization of a set in a heap

The organization of a set in the form of a heap is the basis for the construction of binary search trees. The algorithm for organizing a set in the form of a pile is laborious and requires a high volume of resources. In this paper we have automated the algorithm for organizing a set in the form of a heap. We developed a program in the C++ programming language that implements the heap generation algorithm and I analyzed a numerical example.

The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees

In this paper we consider random trees associated with the genealogy of Crump–Mode–Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump–Mode–Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, and median-of-( $2\ell+1$ ) binary search trees, to name a few, where to our knowledge percolation has not yet been studied.