Let X denote the set of plane trees of n+1 edges with bi-colored leaves, and 2 distinguished internal vertices. Let Y denote the set of plane trees of n edges with bi-colored leaves, and 2 distinguished vertices (not necessarily internal). Then, |X|=|Y|.
Could anyone give a bijection between X and Y?
Note: from "A Combinatorial Identity Concerning Plane Colored Trees and its Applications, Journal of Integer Sequences 20 (2017)", Theorem 6
Bi-colored leaves? Excellent question.
All leaves have many different pigments. Some pigments are used for photoregulations (e.g. phytochromes, blue light receptors, etc), other pigments are used for capturing information (e.g. in vision) other pigments are used to store the absorbed light energy (e.g. in photosynthesis).
The absorbtion and excitation events and the pigment de-excitation events as heat or light emission or energy source in photobiological reactions, are very fast in the order of ns to pc. What counts is how many excited states are able to trigger finally the following dark reactions (de-excitations) leading finally to a biochemical reaction within ms or physiological reactions faster than seconds. Bi-color or not is not the problem. What counts is the superposition of all absorption spectra and excitations in vivo and in situ. This is the primary energy input, which only partially will be stored by the fellow up biochemical reactions. However in bi-colored leaves the anatomy of the leave is mostly very different in the different colored areas, so the photochemical energy storage will be very different as well. WHY and What for? Surviving of the fittest during evolution.
On the higher biological level (e.g. ecology). Bicolored leaves may have a lower photosynthetic energy storage per area, but the "ATTRACTIVITY" (or sexyness and diversity) of being bicolored may differently attract or reject insects or have a protective effect due to camoufflage (tarning and diversion of attacs or even the opposite) as "Carneval-effect" and attractivity...... like using the RED of lip sticks.
That's Biodiversity or Raffinement and optimalisation by selection.
Hi Naduvath,
By bi-colored leaves I mean that every leave is either colored red or green.
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