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36-453: Rensselaer Lake is at the headwaters of the Patroon Creek in the western section of the city of Albany. Once the location of several feeder streams that formed the creek, the land was purchased by the city in 1850 and dammed in 1851 as the city's first municipal source of water. The dam was located roughly six miles (9.7 km) from Albany City Hall , lending the lake its original name of

72-463: A depth-first search and assigning each node's number in postorder . The same numbers may also be generated via a pruning process in which the tree is simplified in a sequence of stages, where in each stage one removes all leaf nodes and all of the paths of degree-one nodes leading to leaves: the Strahler number of a node is the stage at which it would be removed by this process, and the Strahler number of

108-403: A drainage basin are more likely to flood, comparatively, by looking at the separate ratios. Most British rivers have a bifurcation ratio of between 3 and 5. Gleyzer et al. (2004) describe how to compute Strahler stream order values in a GIS application. This algorithm is implemented by RivEX , an ESRI ArcGIS Pro 3.3.x tool. The input to their algorithm is a network of the centre lines of

144-450: A river or stream is the point on each of its tributaries upstream from its mouth / estuary into a lake / sea or its confluence with another river. Each headwater is considered one of the river's sources , as it is the place where surface runoffs from rainwater , meltwater and/or spring water begin accumulating into a more substantial and consistent flow that becomes a first-order tributary of that river. The tributary with

180-636: A tree is the number of stages required to remove all of its nodes. Another equivalent definition of the Strahler number of a tree is that it is the height of the largest complete binary tree that can be homeomorphically embedded into the given tree; the Strahler number of a node in a tree is similarly the height of the largest complete binary tree that can be embedded below that node. Any node with Strahler number i must have at least two descendants with Strahler number i  − 1, at least four descendants with Strahler number i  − 2, etc., and at least 2 leaf descendants. Therefore, in

216-475: A tree is. For each order i in a hierarchy, the i th bifurcation ratio is where n i denotes the number of nodes with order i . The bifurcation ratio of an overall hierarchy may be taken by averaging the bifurcation ratios at different orders. In a complete binary tree, the bifurcation ratio will be 2, while other trees will have larger bifurcation ratios. It is a dimensionless number. The pathwidth of an arbitrary undirected graph G may be defined as

252-406: A tree with n nodes, the largest possible Strahler number is log 2   n  + 1. However, unless the tree forms a complete binary tree its Strahler number will be less than this bound. In an n -node binary tree , chosen uniformly at random among all possible binary trees , the expected index of the root is with high probability very close to log 4   n . In the application of

288-613: A way of measuring the complexity of rivers and streams, by Robert E. Horton  ( 1945 ) and Arthur Newell Strahler  ( 1952 , 1957 ). In this application, they are referred to as the Strahler stream order and are used to define stream size based on a hierarchy of tributaries . The same numbers also arise in the analysis of L-systems and of hierarchical biological structures such as (biological) trees and animal respiratory and circulatory systems, in register allocation for compilation of high-level programming languages and in

324-438: Is marshland . The furthest stream is also often called the head stream. Headwaters are often small streams with cool waters because of shade and recently melted ice or snow. They may also be glacial headwaters, waters formed by the melting of glacial ice . Headwater areas are the upstream areas of a watershed , as opposed to the outflow or discharge of a watershed. The river source is often but not always on or quite near

360-415: Is "in a location that is the farthest, along water miles, from where that river ends." Under this definition, neither a lake (excepting lakes with no inflows) nor a confluence of tributaries can be a true river source, though both often provide the starting point for the portion of a river carrying a single name. For example, National Geographic and virtually every other geographic authority and atlas define

396-470: Is given by Smart. The Strahler numbering may be applied in the statistical analysis of any hierarchical system, not just to rivers. When translating a high-level programming language to assembly language the minimum number of registers required to evaluate an expression tree is exactly its Strahler number. In this context, the Strahler number may also be called the register number . For expression trees that require more registers than are available,

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432-496: The Sethi–Ullman algorithm may be used to translate an expression tree into a sequence of machine instructions that uses the registers as efficiently as possible, minimizing the number of times intermediate values are spilled from registers to main memory and the total number of instructions in the resulting compiled code. Associated with the Strahler numbers of a tree are bifurcation ratios , numbers describing how close to balanced

468-607: The Six Mile Waterworks. Prior to this, the city was served by a private company, the Albany Waterworks Company. The lake fed water to the Bleecker Reservoir where it was used by portions of the city west of Pearl Street to the area around Lark Street . The lake was used as a water supply until 1926, and became a city park under Mayor Erastus Corning II in 1956. In the late 1990s the Six Mile Waterworks

504-405: The Strahler stream order to hydrology, each segment of a stream or river within a river network is treated as a node in a tree, with the next segment downstream as its parent. When two first-order streams come together, they form a second-order stream. When two second-order streams come together, they form a third-order stream. Streams of lower order joining a higher order stream do not change

540-575: The USGS at times considers the Missouri River as a tributary of the Mississippi River . But it also follows the first definition above (along with virtually all other geographic authorities and publications) in using the combined Missouri—lower Mississippi length figure in lists of lengths of rivers around the world. Most rivers have numerous tributaries and change names often; it is customary to regard

576-453: The analysis of social networks . All trees in this context are directed graphs , oriented from the root towards the leaves; in other words, they are arborescences . The degree of a node in a tree is just its number of children. One may assign a Strahler number to all nodes of a tree, in bottom-up order, as follows: The Strahler number of a tree is the number of its root node. Algorithmically , these numbers may be assigned by performing

612-406: The bodies of water, represented as arcs (or edges) joined at nodes. Lake boundaries and river banks should not be used as arcs, as these will generally form a non-tree network with an incorrect topology. Alternative stream ordering systems have been developed by Shreve and Hodgkinson et al. A statistical comparison of Strahler and Shreve systems, together with an analysis of stream/link lengths,

648-562: The confluence of the Madison and Jefferson rivers, rather than the source of its longest tributary (the Jefferson). This contradicts the most common definition, which is, according to a US Army Corps of Engineers official on a USGS site, that "[geographers] generally follow the longest tributary to identify the source of rivers and streams." In the case of the Missouri River, this would have

684-619: The edge of the watershed, or watershed divide. For example, the source of the Colorado River is at the Continental Divide separating the Atlantic Ocean and Pacific Ocean watersheds of North America . A river is considered a linear geographic feature, with only one mouth and one source. For an example, the Mississippi River and Missouri River sources are officially defined as follows: The verb "rise" can be used to express

720-400: The entire 206-acre (83 ha) city park and helped balance the city's budgets for 2002, 2003, and 2004. This sale was criticized as being a fiscal trick , shifting some of the tax burden from residents' property tax bills to their water bills. City property taxes increased that year 7.4 percent, and the water authority also tacked on a 9.5 percent rate increase. Rensselaer Lake is located in

756-451: The general region of a river's source, and is often qualified with an adverbial expression of place. For example: The word "source", when applied to lakes rather than rivers or streams, refers to the lake's inflow . Strahler number In mathematics , the Strahler number or Horton–Strahler number of a mathematical tree is a numerical measure of its branching complexity. These numbers were first developed in hydrology , as

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792-445: The lake has two forks fed by inlet streams; the northern fork by groundwater, and the southern fork by a 1,150-foot (350 m) stream created by two drainage culverts and groundwater seepage. The two forks merge 2,300 feet (700 m) past the southern inlet stream. The lake has a surface area of 35.3 acres (14.3 ha). It has a mean depth of 11 feet (3.4 m) with a maximum depth of over 20 feet (6.1 m). Rensselaer Lake and

828-413: The longest course downstream of the headwaters is regarded as the main stem . The United States Geological Survey (USGS) states that a river's "length may be considered to be the distance from the mouth to the most distant headwater source (irrespective of stream name), or from the mouth to the headwaters of the stream commonly known as the source stream". As an example of the second definition above,

864-551: The longest tributary or stem as the source, regardless of what name that watercourse may carry on local maps and in local usage. This most commonly identified definition of a river source specifically uses the most distant point (along watercourses from the river mouth ) in the drainage basin from which water runs year-around ( perennially ), or, alternatively, as the furthest point from which water could possibly flow ephemerally . The latter definition includes sometimes-dry channels and removes any possible definitions that would have

900-496: The main eastern basin to a shallower western basin which has a depth of 15 feet (4.6 m). Exit 1 and Interstate 90 (I-90) are directly south of the lake and preserve. To the north of the lake and preserve is the town of Guilderland and CSXT railroad tracks. West of the lake is Rapp Road; to the east is Fuller Road and Exit 2 of I-90. The Patroon Creek flows east from the lake towards the Hudson River . The western section of

936-482: The most powerful river, the Amazon , at its mouth). The Ohio River is of order eight and the Mississippi River is of order 10. Estimates are that 80% of the streams on the planet are first to third order headwater streams . If the bifurcation ratio of a river network is high, then there is a higher chance of flooding. There would also be a lower time of concentration. The bifurcation ratio can also show which parts of

972-485: The order of the higher stream. Thus, if a first-order stream joins a second-order stream, it remains a second-order stream. It is not until a second-order stream combines with another second-order stream that it becomes a third-order stream. As with mathematical trees, a segment with index i must be fed by at least 2 different tributaries of index 1. Shreve noted that Horton's and Strahler's Laws should be expected from any topologically random distribution. A later review of

1008-496: The park surrounding it allow fishing, boating, hiking, and picnicking; there is also a playground. The lake is stocked several times a year from the city's current source of water, the Alcove Reservoir . A nature viewing platform allows visitors to view wildlife along a wetland on the lake without disturbing the environment. Two large docks allow for access to the lake by boat. Source (river or stream) The headwater of

1044-467: The relationships confirmed this argument, establishing that, from the properties the laws describe, no conclusion can be drawn to explain the structure or origin of the stream network. To qualify as a stream a hydrological feature must be either recurring or perennial . Recurring (or "intermittent") streams have water in the channel for at least part of the year. The index of a stream or river may range from 1 (a stream with no tributaries) to 12 (globally

1080-628: The river changes names numerous times along its course. However, the source of the Thames in England is traditionally reckoned according to the named river Thames rather than its longer tributary, the Churn — although not without contention. When not listing river lengths, however, alternative definitions may be used. The Missouri River's source is named by some USGS and other federal and state agency sources, following Lewis and Clark 's naming convention, as

1116-547: The river source "move around" from month to month depending on precipitation or ground water levels. This definition, from geographer Andrew Johnston of the Smithsonian Institution , is also used by the National Geographic Society when pinpointing the source of rivers such as the Amazon or Nile . A definition given by the state of Montana agrees, stating that a river source is never a confluence but

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1152-414: The smallest number w such that there exists an interval graph H containing G as a subgraph, with the largest clique in H having w  + 1 vertices. For trees (viewed as undirected graphs by forgetting their orientation and root) the pathwidth differs from the Strahler number, but is closely related to it: in a tree with pathwidth w and Strahler number s , these two numbers are related by

1188-582: The source be well upstream from Lewis and Clark's confluence, "following the Jefferson River to the Beaverhead River to Red Rock River , then Red Rock Creek to Hell Roaring Creek ." Sometimes the source of the most remote tributary may be in an area that is more marsh -like, in which the "uppermost" or most remote section of the marsh would be the true source. For example, the source of the River Tees

1224-617: The source of the Nile River not as Lake Victoria 's outlet where the name "Nile" first appears, which would reduce the Nile's length by over 900 km (560 mi) (dropping it to fourth or fifth on the list of world's rivers), but instead use the source of the largest river flowing into the lake, the Kagera River . Likewise, the source of the Amazon River has been determined this way, even though

1260-587: The western section of the city of Albany, with a portion stretching into the neighboring town of Guilderland . It is the namesake and central feature of the Albany Pine Bush Rensselaer Lake Preserve and Park, also known as the Six Mile Waterworks. It is a part of the Albany Pine Bush Preserve. The Adirondack Northway ( I-87 ) is carried over a narrow channel of the lake by a bridge just north of Exit 1. The channel connects

1296-618: Was restored after decades of disrepair. The city partnered with the Albany Pine Bush Preserve Commission, area schools, businesses, youth organizations and individual sponsors to build new docks, a boardwalk, and a nature trail. An open-air interpretive center is planned. In 2003, the Albany Water Authority paid the city of Albany $ 7 million for a 50-year lease of the Six Mile Waterworks and Rensselaer Lake for use as an emergency backup water supply. This included

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