# A Tree Leaf Model, Part 2

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David A. Tanzer, December 29, 2020

## A leaf as a tree of pipes

Last time we set out the model of a leaf as a set of square cells in the plane. But there’s more structure to be defined: the veins.

All the veins taken together make up a ‘transport system’ for pumping fluid from the root to each of the cells.

Biological note: The cells need water to do their work of converting sunlight into energy-bearing molecules. Another transport system, going the other way, distributes these molecules to the rest of the plant. In Xia’s model, only the transport system from the root is considered.

Here, the transport system will be modeled as system of ‘pipes’ which connect the cells (squares in the grid). Every pipe goes from a source cell to a target cell.

The pipes have varying lengths and thicknesses. Thicker pipes are needed for parts of the transport system where a greater flow rate is called for. The pipes that lead from the root cell will be the thickest, as the root must supply enough fluid to reach all the cells in the leaf. This does agree with nature, where the thickest veins are indeed the ones at the “root” of the leaf.

Now the transport system is not just any old collection of pipes – for example, we would never expect to see a circular chain of pipes in the system. The pipes are organized into a branching structure that starts at the root and radiates outward to reach all of the cells.

Mathematically, it is a graph where edges are the pipes and the nodes are the cells. As we said, this graph will never contain any circular paths. More specifically, the type of graph is called a tree, which means that every cell has exactly one parent cell that pipes into it.

Exception: the root of the leaf has no parent cell, as it is treated as it is modeled as the originating source of all the fluid. (In a fuller model that includes more of the plant, the root of the leaf would be supplied by a pipe coming from somewhere in the stem.)

And every cell has a set of child cells, which it has pipes to. The cells at the fringe of the leaf have no children.

Starting at any cell, we can find its parent, then the grandparent, etc. This gives a chain of cells leading back to the root, with a pipes between successive cells in this chain. Put together, these make a chain of pipes going from the root to the cell.

Every cell has a pipeline leading to it, but due to the branching structure, groups of pipelines may share overlapping segments.

Pipes closer to the root are shared among more pipelines. Such pipes will supply a larger number of cells downstream. Hence a greater flow rate is needed through these pipes, so they will be wider.