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Section 4:   An Exploration of the Divide Tree

The Divide Tree is a simplified model of a Ridge Network that provides only the data points useful to Prominence Theory.  An idealized Divide Tree would include only the summits and saddles above a threshold prominence value, and the ridges that form the critical paths between them.

The Divide Tree is a dendritic model with no closed loops.  Basin saddles and the ridges that rise from them are removed from the tree, as they have no application to prominence.  The series of prominence maps at provide many examples of the Divide Tree form.

4.1        Does the Divide Tree have a root?

Divide trees have a root, which is represented by the highest point on the surface, and branches that connect to N number of summits at the terminus of various minor divides.   The root high point, being a function of the geomorphology of the terrain, may be located anywhere on the tree.  For example, Mt. McKinley appears at the Northern end of the North American divide tree, and not on the continental divide.   Divide trees, like a natural tree, form major and minor branches. 

The backbone of the American divide tree is the Continental Divide, stretching from the Bering Sea to the Straits of Magellan.  The continental divide is a construct more central to hydrology than to the Divide Tree.  It stems from a notion of a continent with two oceans (such as the Americas) and thus with two systems of drainage.  In the Americas, the continental divide mainly coincides with the crest of the continents' major mountain system (except in Alaska), and thus serves multiple purposes in the public imagination.

Secondary divides can also be inferred from hydrology, at the separation of major drainages whose waters do not merge. North America's secondary divides separate watersheds of the Arctic Ocean, Hudson Bay, Saint Lawrence, Atlantic Seaboard, Mississippi, Rio Grande, Colorado, Columbia, Yukon, etc.

What is needed is a hierarchy of arterial and branch divides that would be derived purely from topological form and not from the outlying hydrological network.  Continental and secondary divides (in the above sense) are based on the coincidence of oceanic shape, not on a characteristic of the divide tree itself. Possible solutions would be to assign weight to branches of the divide tree based on numbers of summits, based on the prominence values of major summits, or based on the mass or regions of prominence islands or domains (described in a later section.)   The author has not attempted to assign weighting to the branches of a divide tree, though this would be an informative mathematical study.

4.2        Automatic extraction of Prominence

Edward Earl, an avid mountain climber, engineer, and mathematician has extensively studied the nature of prominence and the divide tree.  Over the course of several years, Earl has developed an impressive program named Winprom that extracts prominence data from digital elevation models. 

Winprom automatically generates a Divide Tree by searching for saddles and summits.  Like Maxwell before him, Earl (independently) realized that the key to finding the critical path between saddle and summit is to begin with saddles.  Whereas a summit has infinite descending slope lines, most of which join together on slope or channel, a saddle has just two ascending slope lines, which always reach a summit.   Figure X shows a detail of Winprom output data.  Note that summits are denoted with a yellow triangle, and saddles with a green circle.  Critical paths from saddle to summit are denoted by straight-lines.

4.3        Highly Connected Peaks

With the position of all summits established on a Divide Tree, we can detect a simple typology of mountains based on their number of saddle connections. 

The direction from summit to key saddle can lead toward or lead away from (be in opposition to) the bulk of the continental tree.  Most summits attach to the bulk of a divide tree across the summit.   Mt. Elbert, the high point of the Rocky Mountains is an example of a summit in opposition to the rest of the continent.  The Elbert KS is in the direction of Mt. Whitney, the high point of the Sierra Nevada, which is away from the rest of North America.

In addition to the key saddle, many summits have additional saddles that also connect along the divide tree to higher ground.  So-called Secondary Saddles are always the key saddles of higher mountains, and are always lower than the summit's key saddle.  Mount Elbert's "prominence island" is the large highland area (above 5,360' elevation) of the Rocky Mountains and the Great Basin of Nevada and Utah.  In addition to the Elbert KS in eastern California,  two secondary saddles also connect Mt. Elbert to higher ground in either direction along the continental divide.  The Whitney KS separates Mt. Elbert from the Mexican Volcanoes to the south.  The Orizaba KS in northern Canada separates Mt. Elbert from the high mountains of the Yukon and Alaska to the north.

It turns out that more than half of the high-prominence mountains have just one connection, and are thus "terminal" summits, being on the terminus of a branch of the divide tree relative to their elevation.  Terminal summits seem to occur in all terrain types, at all elevations and levels of prominence.  It is not known how their frequency changes with different prominence values.  Of the P>5,000' summits in the contiguous United States, 40 of 57 are terminal, 11 have two connections, 4 have three connections and 2 have four connections.  Presumably mountains close to major divides will tend to multiple connections; therefore connectivity may be some measure of the centrality of a given divide. 

The two quadruple connections in the Top 57 are Gannett Peak WY, and North Schell Peak, NV.  Correspondence with a fellow researcher in Britain suggests that mountains with five and six saddles occur with great rarity, but the author has not searched for these geographical oddities.

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