On locally finite ordered rooted trees and their rooted subtrees
Abstract
In this article we compare the known dynamical polynomial time algorithm for the game-over attack strategy, to that of the brute force approach; of checking all the ordered rooted subtrees of a given tree that represents a given computer network. Our approach is purely enumerative and combinatorial in nature. We first revisit known results about a doubly exponential sequence and generalize them. We then consider both finite and locally finite ordered rooted trees (LFOR-trees), and the class of their finite ordered rooted subtrees of bounded height, describing completely the LFOR-trees with no leaves where the number of ordered rooted subtrees of height at most $h$ are bounded by a polynomial in $h$. We finally consider general LFOR-trees where each level can have leaves and determine conditions for the number of ordered rooted subtrees of height at most $h$ to be bounded by a polynomial in $h$.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2023
- DOI:
- arXiv:
- arXiv:2312.11379
- Bibcode:
- 2023arXiv231211379A
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 35 pages, 7 figures