Abstract
This study of tree morphology is presented in three parts. Part 1 deals with the over-all shape of trees (their profiles). A mathematical model based on the distribution of leaves shows that all profiles, from rounded to conical, is determined by one parameter. Part 2 deals with the angle of branching. Field measurements show that the sum of the areas of stems exiting a fork is usually greater than the area of the stem entering the fork. Mathematical analysis shows that this “bulking up” actually reduces the quantity of plant tissue incorporated in the branching. Furthermore, the angle of branching increases with the degree of bulking up. Part 3 brings together the concepts of the first two parts: here, the cross-sectional area of the trunk as a function of position along the trunk is determined by the profile of the tree and the angle of side branching off the trunk. From field observation, the profile of a tree (i.e., the outline of its crown) has two prominent characteristics: (i) azimuthal symmetry about the central axis (often the main stem or trunk), as evidenced both in foliage and scaffolding; and (ii) decrease in leaf density from branch-end toward the central axis. From these two conditions (including a presumed radial distribution of leaves) a mathematical model is developed using the calculus of variations that predicts the profile The results of this analysis are consistent with the general observation that profiles range from the nearly spherical in the case of uniform distribution of leaves throughout the crown, to essentially conical when the leaves are found largely on the branch-ends. The results are presented in figures showing theoretical profiles overlaid on photographs of representative trees. Part 2 is based on field measurements that show that the cross-sectional area of a branch (or stem) entering a fork (in the direction of water transport) is less than the sum of the cross-sectional areas of the branches leaving that fork. Again, using the calculus of variations, it is shown that the angle of branching can be related to this “bulking up”. Field measurements are in general agreement with this prediction. Part 3 provides a separate test of the validity of the concepts introduced in the first two Parts. Using equations appropriate to a tree with a single main stem and horizontal side branches, the crosssectional area of the trunk is calculated as a function of position on the trunk. The results are compared with field measurements.
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Claims 10
Claim · model-written summary
A leaf distribution model describes tree profiles with one parameter.
Verbatim
e0“all profiles, from rounded to conical, is determined by one parameter”
t0“Tree Shape”
Claim · model-written summary
The total area of stems leaving a fork usually exceeds the entering stem area.
- Outcome
- the sum of the areas of stems exiting a forke0
- Direction
- greater thane0
- Comparator
- the area of the stem entering the forke0
Not reportedTaxon
Verbatim
e0“the sum of the areas of stems exiting a fork is usually greater than the area of the stem entering the fork”
Claim · model-written summary
The model finds that bulking up reduces branching tissue quantity.
- Outcome
- quantity of plant tissue incorporated in the branchinge0
- Direction
- reducese0
- Qualifier
- this “bulking up”e0
Not reportedTaxon
Verbatim
e0“this “bulking up” actually reduces the quantity of plant tissue incorporated in the branching”
Claim · model-written summary
Branching angle increases with the degree of bulking up.
Not reportedTaxonComparator
Verbatim
e0“the angle of branching increases with the degree of bulking up”
Claim · model-written summary
Trunk cross-sectional area is determined by tree profile and side branch angle.
- Taxon
- the treee0
- Organ / stage
- trunke0
- Outcome
- the cross-sectional area of the trunke0
- Qualifier
- the profile of the tree and the angle of side branching off the trunke0
Not reportedDirection
Verbatim
e0“the cross-sectional area of the trunk as a function of position along the trunk is determined by the profile of the tree and the angle of side branching off the trunk”
Claim · model-written summary
Observed crowns show azimuthal symmetry.
Not reportedTaxonDirection
Verbatim
e0“azimuthal symmetry about the central axis (often the main stem or trunk), as evidenced both in foliage and scaffolding”
Claim · model-written summary
Leaf density decreases from branch ends toward the central axis.
Not reportedTaxon
Verbatim
e0“decrease in leaf density from branch-end toward the central axis”
Claim · model-written summary
Uniform crown leaf distribution produces nearly spherical predicted profiles.
Not reportedTaxonDirection
Verbatim
e0“profiles range from the nearly spherical in the case of uniform distribution of leaves throughout the crown”
Claim · model-written summary
Leaves concentrated at branch ends produce essentially conical predicted profiles.
Not reportedTaxonDirection
Verbatim
e0“to essentially conical when the leaves are found largely on the branch-ends”
Claim · model-written summary
Field branching measurements generally agree with the predicted relationship.
Not reportedTaxonDirection
Verbatim
e0“Field measurements are in general agreement with this prediction”