File:Nested set V4; trees and subtrees with open nodes.svg

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Summary

Description

A big tree with small trees as nodes and leaves
The leaves are elements of the nested set V4 - and therefore correspond to integers 0...15

The leaves are the trees from this file, and correspond to nested sets.


This tree refines the concept of weight of a binary number:

  • The root is the set of all non-negative integers.
  • The direct children of the root are the sets of integers with a certain binary weight:
Knot 1 is the set of integers with weight 1 (i.e. powers of 2), knot 2 the set of integers with weight 2, etc.
  • The knots in the next row specify the weight of n in the n-th digit:
The first non-leaf knot (black 1 and white 1) is the set of Fermat powers of 2, i.e. integers of the form 2^2^n.

The 16 leaves are 4-bit binary numbers, and the number of knots including root and leaves is 43.

The number of knots for n-bit binary numbers with n=0..4 is 2,4,9,20,43 (possibly Sloane's A123720).
Source Own work
Author
Watchduck
You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".

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Date/TimeThumbnailDimensionsUserComment
current17:53, 4 October 2015Thumbnail for version as of 17:53, 4 October 20154,052 × 1,233 (743 KB)Watchduckadd integers to the leaves
16:46, 20 November 2014Thumbnail for version as of 16:46, 20 November 20144,052 × 1,193 (576 KB)Watchduck{{Information |Description ={{en|1=a big tree with the elements of the nested set V<sub>4</sub> as leaves (represented by trees as in File:Nested set V4; circles and trees.svg) the other knots do not stand for int...

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