Instructions
The file catalogn.dat (n=2,...,14) includes all trees with n edges.
(change INSTRUCTIONS.txt in the address of this page to catalogn.dat)
The file treen.pdf gives pictures of all trees with n edges, with the
"passport" written at the top of the picture.
(change INSTRUCTIONS.txt in the address of this page to treen.pdf,
n=2,...,14)
The files with pictures of trees with ten edges is split into three:
tree10a.pdf, tree10b.pdf and tree10c.pdf.
Trees with more than 10 edges are too big to include
here, so we drew some of the pictures for 11, 12, 13 edges in
the files tree11a.pdf, tree12a.pdf, tree13a.pdf.
The catalogs, though, are complete through 14 edges.
The number of edges of the tree equals the
degree of the corresponding shabat polynomial, and is one fewer than
the number of vertices.
The catalog shows for each tree, the corresponding matching of edges
(counterclockwise labelling) together with all rotations (so that you
can search for a particular description). This is followed by the
vertices, the value of the shabat poly, and the order of the vertex.
Finally, x,y coordinates of points on the tree are given so that a
picture can be drawn by connecting the points with line segments. The
tree is traversed twice so simply connecting consecutive points with
line segments will give a proper picture.
To find a particular tree, draw it on a piece of
paper, then label the edges counterclockwise from 1 to 2n where
n= is the number of edges (so each edge is counted twice), starting
from any edge.
Then write down one row of numbers from 1 to 2n and underneath these numbers
write the number of the matching interval. So if an edge has labels 2
and 5 then underneath 2 write 5 and underneath 5 write 2.
This second row is the matching for the shabat polynomial and tree.
--If your editor can handle large files:
Use your editor to find this matching in the file catalog-n.dat.
Note that each integer together with the space(s) before it take a total
of three spaces. So search for 4 3 12 1 not 4 3 12 1.
You will see a block of matchings which include your matching and all
its rotations, which correspond to simply starting with a different
edge labelled as 1.
If you have execute privileges on this machine,
--type the (windows) command
search 8 3 2 5 4 7 6 1
and it will find the tree information (only requires at least one
space between each integer) for this matching.
This uses the program sed.exe from UnxUtil (please read the
GNU-License in the file gpl-3.0.txt).
The output will contain some of the rotations of the matching, the
vertices, value and order, and the tree. Not all rotations will be
included since it is a string matching program that gives everything
from the matching to the end of the tree. A similar batch command can
be written for unix systems (just imitate the search.cmd command using
the normal sed).
REMARK: we only outputted 6 digits of the points on the branches to
draw the pictures. We could easily have written out many more digits.
Trees of much higher degree can be easily computed individually
as well. The barrier to including them in the catalog is their shear number.
Increasing the number of edges by 1 increases the number of trees by
a factor of about 4.
The file time-errors.txt gives the number of trees of each degree, the
time it took to compute and the max errors in computing the vertices
of the tree. The matchings for trees are generated by modifying
matchings for lower degree trees, so the matchings for these timing
results had to be regenerated from scratch each time. In other
words, the code would compute matchings for all degree 2-12 vertices
faster in one run than it took to do them separately. The conformal
mapping program takes just a matching for a tree as input, it does not
depend on lower degree matchings. Roughly there are three times
as many trees if the number of vertices is increased by 1, so it takes
roughly four times as long because of the duplication, if run
separately.
The file errors.txt gives the 7 trees of degree 14 where the Newton's
method used to find the branches of the tree failed to converge at a
point. Newton's method converged for all vertices of all degree 14
trees. It failed only in 7 cases for a point on a branch near a degree 2 vertex
where the value of the approximate polynomial was too small (or too
close to 1) to locate
a nearby non-vertex point on the correct branch.
This does not affect the
catalog, only the pictures of these trees. This problem can be avoided
by increasing the number of subdivisions (12 was chosen initially).
The tree can be drawn in any case by omitting this point or by using
the value of the polynomial at the nearby vertex instead of a
corresponding point on the branch.