\name{spantree}
\alias{spantree}
\alias{cophenetic.spantree}
\alias{as.hclust.spantree}
\alias{plot.spantree}
\alias{lines.spantree}
\alias{spandepth}

\title{Minimum Spanning Tree}
\description{
  Function \code{spantree} finds a minimum spanning tree
  connecting all points, but disregarding dissimilarities that are at or
  above the threshold or \code{NA}.  
}
\usage{
spantree(d, toolong = 0)
\method{as.hclust}{spantree}(x, ...)
\method{cophenetic}{spantree}(x)
spandepth(x)
\method{plot}{spantree}(x, ord, cex = 0.7, type = "p", labels, dlim,
     FUN = sammon,  ...)
\method{lines}{spantree}(x, ord, display="sites", ...)
}

\arguments{
  \item{d}{Dissimilarity data inheriting from class \code{dist} or
    a an object, such as a matrix, that can be converted to a
    dissimilarity matrix. Functions \code{\link{vegdist}} and
    \code{\link{dist}} are some functions producing suitable
    dissimilarity data.}
  \item{toolong}{ Shortest dissimilarity regarded as \code{NA}.
    The function uses a fuzz factor, so
    that dissimilarities close to the limit will be made \code{NA}, too.
    If \code{toolong = 0} (or negative), no dissimilarity is regarded
    as too long.
  }
  \item{x}{A \code{spantree} result object.}
  \item{ord}{An ordination configuration, or an ordination result known
    by \code{\link{scores}}.}
  \item{cex}{Character expansion factor.}
  \item{type}{Observations are plotted as points with
    \code{type="p"} or \code{type="b"}, or as text label with
    \code{type="t"}. The tree (lines) will always be plotted.}
  \item{labels}{Text used with \code{type="t"} or node names if this is
    missing.}
  \item{dlim}{A ceiling value used to highest \code{cophenetic} dissimilarity.}
  \item{FUN}{Ordination function to find the configuration from
    cophenetic dissimilarities. }
  \item{display}{Type of \code{\link{scores}} used for \code{ord}.}
  \item{\dots}{Other parameters passed to functions.}
}
\details{
  
  Function \code{spantree} finds a minimum spanning tree for
  dissimilarities (there may be several minimum spanning trees, but the
  function finds only one). Dissimilarities at or above the threshold
  \code{toolong} and \code{NA}s are disregarded, and the spanning tree
  is found through other dissimilarities. If the data are disconnected,
  the function will return a disconnected tree (or a forest), and the
  corresponding link is \code{NA}. Connected subtrees can be identified
  using \code{\link{distconnected}}.

  Minimum spanning tree is closesly related to single linkage
  clustering, a.k.a. nearest neighbour clustering, and in genetics as
  neighbour joining tree available in \code{\link{hclust}} and
  \code{\link[cluster]{agnes}} functions. The most important practical
  difference is that minimum spanning tree has no concept of cluster
  membership, but always joins individual points to each other. Function
  \code{as.hclust} can change the \code{spantree} result into a
  corresponding \code{\link{hclust}} object.

  Function \code{cophenetic} finds distances between all points along
  the tree segments. Function \code{spandepth} returns the depth of
  each node. The nodes of a tree are either leaves (with one link) or
  internal nodes (more than one link). The leaves are recursively
  removed from the tree, and the depth is the layer at with the leaf
  was removed. In disconnected \code{spantree} object (in a forest)
  each tree is analysed separately and disconnected nodes not in any
  tree have depth zero.
  
  Function \code{plot} displays the tree over a
  supplied ordination configuration, and \code{lines} adds a spanning
  tree to an ordination graph. If configuration is not supplied for \code{plot},
  the function ordinates the cophenetic dissimilarities of the
  spanning tree and overlays the tree on this result. The default
  ordination function is \code{\link[MASS]{sammon}} (package \pkg{MASS}),
  because Sammon scaling emphasizes structure in the neighbourhood of
  nodes and may be able to beautifully represent the tree (you may need
  to set \code{dlim}, and sometimes the results will remain
  twisted). These ordination methods do not work with disconnected
  trees, but you must supply the ordination configuration. Function
  \code{lines} will overlay the tree in an existing plot. 

  Function \code{spantree} uses Prim's method
  implemented as priority-first search for dense graphs (Sedgewick
  1990). Function \code{cophenetic} uses function
  \code{\link{stepacross}} with option \code{path = "extended"}. The
  \code{spantree} is very fast, but \code{cophenetic} is slow in very
  large data sets. 
}
\value{
  Function \code{spantree}
  returns an object of class \code{spantree} which is a
  list with two vectors, each of length \eqn{n-1}. The
  number of links in a tree is one less the number of observations, and
  the first item is omitted. The items are  
  \item{kid }{The child node of the parent, starting from parent number
    two. If there is no link from the parent, value will be \code{NA}
    and tree is disconnected at the node.}
  \item{dist }{Corresponding distance. If \code{kid = NA}, then
    \code{dist = 0}.}
  \item{labels }{Names of nodes as found from the input dissimilarities.} 
  \item{call}{The function call.}
}
\references{
 Sedgewick, R. (1990). \emph{Algorithms in C}. Addison Wesley. 
}
\author{ Jari Oksanen }
\note{
  In principle, minimum spanning tree is equivalent to single linkage
  clustering that can be performed using \code{\link{hclust}} or
  \code{\link[cluster]{agnes}}. However, these functions combine
  clusters to each other and the information of the actually connected points
  (the ``single link'') cannot be recovered from the result. The
  graphical output of a single linkage clustering plotted with
  \code{\link{ordicluster}} will look very different from an equivalent
  spanning tree plotted with \code{lines.spantree}.
}


\seealso{\code{\link{vegdist}} or \code{\link{dist}} for getting
  dissimilarities,  and \code{\link{hclust}} or
    \code{\link[cluster]{agnes}} for single linkage clustering. 
}
\examples{
data(dune)
dis <- vegdist(dune)
tr <- spantree(dis)
## Add tree to a metric scaling 
plot(tr, cmdscale(dis), type = "t")
## Find a configuration to display the tree neatly
plot(tr, type = "t")
## Depths of nodes
depths <- spandepth(tr)
plot(tr, type = "t", label = depths)
## Plot as a dendrogram
plot(as.hclust(tr))
}
\keyword{ multivariate}

