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```
#' Calculation of weighted l1 distance index for rooted binary trees
#'
#' This function calculates the weighted l1 distance index \eqn{D_{l1}(T)}{Dl1(T)} for a
#' given rooted binary tree \eqn{T}. \eqn{D_{l1}(T)}{Dl1(T)} is defined as
#' \deqn{D_{l1}(T)=\sum_{z=2}^n z \cdot |f_n(z)-p_n(z)|}{Dl1(T)= \sum z*|f_n(z)-p_n(z)|
#' over all possible sizes 2<=z<=n} in which \eqn{n} denotes the
#' number of leaves of \eqn{T}, \eqn{f_n(z)} denotes the frequency of pending subtrees
#' of size \eqn{z} in \eqn{T} and \eqn{p_n(z)} is the expected number of
#' pending subtrees of size \eqn{z} under the Yule model, i.e. \eqn{p_n(z)=\frac{1}{n-1}}{p_n(z)=1/(n-1)}
#' if \eqn{z=n} and otherwise \eqn{\frac{n}{n-1}\cdot\frac{2}{z\cdot(z+1)}}{n/(n-1)*2/(z*(z+1))}.\cr\cr
#' For \eqn{n=1} the function returns \eqn{D_{l1}(T)=0}{Dl1(T)=0}.
#'
#' @param tree A rooted binary tree in phylo format.
#'
#' @return \code{weighL1distI} returns the weighted l1 distance index of the given tree.
#'
#' @author Sophie Kersting
#'
#' @references M. G. Blum and O. François. On statistical tests of phylogenetic tree imbalance: The Sackin and other indices revisited. Mathematical Biosciences, 195(2):141-153, 2005. doi: 10.1016/j.mbs.2005.03.003.
#'
#' @examples
#' tree <- ape::read.tree(text="((((,),),(,)),(((,),),(,)));")
#' weighL1dist(tree)
#'
#'@export
weighL1dist <- function(tree)
{
#Check for errors in input
if (!is_binary(tree)) stop("The input tree must be binary.")
n <- length(tree$tip.label)
if(n == 1) {return(0)}
# get the size of each subtree
sub.size <- get.subtreesize(tree=tree)
# summarise over all subtree sizes
wL1d_val <- 0
for(z in 2:n)
{
f_z <- sum(sub.size==z) / tree$Nnode
if(z == n) {
p_z <- 1/(n-1)
} else {
p_z <- n/(n-1)*2/(z*(z+1))
}
wL1d_val <- wL1d_val + z*abs(f_z-p_z)
}
return (wL1d_val)
}
```

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