Tree Model and Five Types of Selection

Let $${GP}$$ be a set of indirected arcs. Number the two edges of each arcs with natural numbers. If one edge is numbered $${i}$$, let the other edge be numbered $${i + 1}$$ or $${i - 1}$$ ($${1 \leqq i - 1}$$). Let the arc with the edges numbered $${i}$$ and $${i + 1}$$ be denoted as $${\gamma_{\alpha_i i}}$$. Let the edge numbered $${i}$$ of $${\gamma_{\alpha_i i}}$$ be denoted as $${i_{\alpha_{i - 1}}}$$.

Let one $${\gamma_{\alpha_i i}}$$ correspond to at least one $${\gamma_{\alpha_{i - 1} i - 1}}$$ and let one $${i_{\alpha_{i - 1}}}$$ correspond to at least one $${\gamma_{\alpha_{i - 1} i - 1}}$$ to form a graph $${\gamma}$$. Then the graph $${\gamma}$$ would resemble a tree. We call $${\gamma}$$ a $${\textit{tree model}}$$.

In the tree model $${\gamma}$$, $${i_{\alpha_{i - 1}}}$$ is a multifurcation from which arcs $${\gamma_{\alpha_i i}}$$, $${\gamma_{\alpha_i' i}}$$, $${\cdots}$$ emerge towards the next multifurcations numbered $${i + 1}$$.

Let the set of arcs with two edges numbered $${i}$$ and $${i + 1}$$ be denoted as follows:

$$
GP_i \equiv \{ \, \gamma_{\alpha_i i} \, \}_{\alpha_i \in A_i} \subset GP
$$

where

$$
GP \equiv \{ \, \gamma_{\alpha_i i} \, \}_{\alpha_i \in A}
$$

Among $${GP_i}$$, let the set of arcs such that the edge numbered $${i}$$ is $${i_{\alpha_{i - 1}}}$$ be denoted as follows:

$$
GP_{i_{\alpha_{i - 1}}} \equiv \{ \, \gamma_{\alpha_i i} \, \}_{\alpha_i \in A_{i_{\alpha_{i - 1}}}} \subset GP_i \subset GP
$$

In a tree model $${\gamma}$$, let

$$
m \equiv \#GP_{i_{\alpha_{i -1}}}
$$

Let the state of $${\gamma_{\alpha_i i}}$$ (or $${\gamma_{\alpha_{i - 1} i - 1}}$$) be denoted as $${\psi_{\alpha_i i}}$$ (or $${\psi_{\alpha_{i - 1} i - 1}}$$), then selection appear to be classified into the following five types:

$${(\mathrm{i})}$$  $${\textit{Deterministic selection}}$$. If there appear to be $${m}$$ arcs in front of an individual $${j}$$ at $${i_{\alpha_{i - 1}}}$$, but actually, the selection is already determined by the fate of $${j}$$, by certaion laws of classical mechanics, or for some other reasons, and if the next arc to be selected is $${\gamma_{\alpha_i i}}$$ whose probability of being chosen from $${GP_{i_{\alpha_{i -1}}}}$$ is $${1}$$, and the probabilities for the other arcs in $${GP_{i_{\alpha_{i -1}}}}$$ are $${0}$$, where

$$
\psi_{\alpha_{i - 1} i-1} = \psi_{\alpha_i i}
$$

then let the selection of $${\gamma_{\alpha_i i}}$$ among $${GP_{i_{\alpha_{i -1}}}}$$ at $${i_{\alpha_{i - 1}}}$$ be called a $${\textit{deterministic selection}}$$.

$${(\mathrm{ii})}$$  $${\textit{Probabilistic selection}}$$. Let the probability that $${\gamma_{\alpha_i i}}$$ is selected from $${GP_{i_{\alpha_{i -1}}}}$$ be denoted as $${p_{\alpha_i i}}$$ where

$$
0 \leqq p_{\alpha_i i} \leqq 1
$$

and

$$
\sum_{\alpha_i \in A_{i_{\alpha_{i - 1}}}} p_{\alpha_i i} = 1
$$

hold. If

$$
\psi_{\alpha_{i - 1} i-1} = \sum_{\alpha_i \in A_{i_{\alpha_{i - 1}}}} p_{\alpha_i i} \psi_{\alpha_i i}
$$

then let the selection of $${\gamma_{\alpha_i i}}$$ among $${GP_{i_{\alpha_{i -1}}}}$$ at $${i_{\alpha_{i - 1}}}$$ be called a $${\textit{probabilistic selection}}$$.

$${(\mathrm{iii})}$$  $${\textit{Deliberate selection}}$$. Let each $${a_{\alpha_i i} \in \mathbb{R}}$$ be a real number that $${j}$$ chose at $${j}$$'s own will, according to $${j}$$'s preferences, or for some other reasons, or freely without any reasons, instead of $${p_{\alpha_i i}}$$. We call each $${a_{\alpha_i i}}$$ the $${\textit{weight}}$$ of $${\gamma_{\alpha_i i}}$$. If

$$
\psi_{\alpha_{i - 1} i-1} = \sum_{\alpha_i \in A_{i_{\alpha_{i - 1}}}} a_{\alpha_i i} \psi_{\alpha_i i}
$$

then let the selection of $${\gamma_{\alpha_i i}}$$ among $${GP_{i_{\alpha_{i -1}}}}$$ at $${i_{\alpha_{i - 1}}}$$ be called a $${\textit{deliberate selection}}$$.

$${(\mathrm{iv})}$$  $${\textit{Arbitrary selection}}$$. Let each $${\omega_{\alpha_i i} \in \mathbb{R}}$$ be any real number independent of $${j}$$, used instead of $${p_{\alpha_i i}}$$ or $${a_{\alpha_i i}}$$. We also call each $${\omega_{\alpha_i i}}$$ the $${\textit{weight}}$$ of $${\gamma_{\alpha_i i}}$$. If

$$
\psi_{\alpha_{i - 1} i-1} = \sum_{\alpha_i \in A_{i_{\alpha_{i - 1}}}} \omega_{\alpha_i i} \psi_{\alpha_i i}
$$

then let the selection of $${\gamma_{\alpha_i i}}$$ among $${GP_{i_{\alpha_{i -1}}}}$$ at $${i_{\alpha_{i - 1}}}$$ be called an $${\textit{arbitrary selection}}$$.

$${(\mathrm{v})}$$  $${\textit{Others}}$$.

Let the number of times each arc $${\gamma_{\alpha_i i}}$$ is taken out of $${GP_{i_{\alpha_{i - 1}}}}$$ at $${i_{\alpha_{i - 1}}}$$ be denoted as $${n(i)}$$ ($${1 \leqq i \leqq m}$$).

The apparent difference among the five types of selection $${(\mathrm{i})}$$ to $${(\mathrm{v})}$$ is solely due to the finiteness of each $${n(i)}$$. In the case where $${n(i) = \infty}$$ for all $${i}$$ ($${1 \leqq i \leqq m}$$), all five types of selection $${(\mathrm{i})}$$ to $${(\mathrm{v})}$$ are equivalent to probabilistic selection (assuming that probabilistic selection includes deterministic selection). On the other hand, the concept of probability is defined only in the case where $${n(i) = \infty}$$ for all $${i}$$ ($${1 \leqq i \leqq m}$$). Therefore, in the case where $${n(i) < \infty}$$ for all $${i}$$ ($${1 \leqq i \leqq m}$$), selection cannot be classified into the five types $${(\mathrm{i})}$$ to $${(\mathrm{v})}$$, but instead into four types: $${(\mathrm{i})}$$, $${(\mathrm{iii})}$$, $${(\mathrm{iv})}$$, $${(\mathrm{v})}$$.