## Trees

Using a tree graph, we can create a circulation pattern of n organs or points in which a volume of blood can travel from any of the n points to any other point. The tree graph pattern in the bee illustrates blood flow from the heart to the brain through the dorsal vessel and aorta. This graph will be useful when we consider open circulations in which materials do not necessarily pass through the heart each time before going to a new destination.

If we must shrink a closed system for economic or anatomical reasons minimizing tubes and distances, then our graph having the n points as vertices and the connectors as edges forms a tree graph. The problem now is to find an efficient algorithm for deciding which of the n(n-2) possible trees connecting all our points traverses the least distance and, consequently, uses the least materials or energy. Evolution has created closed circulations that do this. Solutions assume we can measure all distances. Again, we can formulate the problem using a weighted algorithm and what's known as the greedy algorithm to find the optimal solutions.

A greedy or single-minded algorithm performs a single step over and over until the steps can no longer be repeated. The algorithm then chooses the next step and continues until stoppage. Results may not be perfect but are a first approximation. We have no algorithm, for example, that generates a pattern of coloring using the fewest colors for every map, but if we color in as many regions as possible with one color before choosing the next color, and then choose a new color only after we have exhausted the last, and we repeat this sequence until all patches are colored, we have followed a greedy algorithm.

At one point in the process a local optimum becomes a global optimum. Similar logic helps find a dominating set or the group of vertices having all other vertices in the graph as its neighbors. One starts with the vertex having the most vertices in the graph linking to it and then chooses the vertices of the next largest degree and so on until we find the dominating set. A hemocoel containing declining volumes of hemolymph passes through states in which local deep pools grow shallower progressing to global behavior as the volume of hemolymph becomes a two-dimensional film wetting all surfaces (Ref: Greedy Algorithm).

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