code: def cophenet(Z, Y=None):
      """
      Calculate the cophenetic distances between each observation in
      the hierarchical clustering defined by the linkage ``Z``.
  
      Suppose ``p`` and ``q`` are original observations in
      disjoint clusters ``s`` and ``t``, respectively and
      ``s`` and ``t`` are joined by a direct parent cluster
      ``u``. The cophenetic distance between observations
      ``i`` and ``j`` is simply the distance between
      clusters ``s`` and ``t``.
  
      Parameters
      ----------
      Z : ndarray
          The hierarchical clustering encoded as an array
          (see `linkage` function).
      Y : ndarray (optional)
          Calculates the cophenetic correlation coefficient ``c`` of a
          hierarchical clustering defined by the linkage matrix `Z`
          of a set of :math:`n` observations in :math:`m`
          dimensions. `Y` is the condensed distance matrix from which
          `Z` was generated.
  
      Returns
      -------
      c : ndarray
          The cophentic correlation distance (if ``Y`` is passed).
      d : ndarray
          The cophenetic distance matrix in condensed form. The
          :math:`ij` th entry is the cophenetic distance between
          original observations :math:`i` and :math:`j`.
  
      See Also
      --------
      linkage : for a description of what a linkage matrix is.
      scipy.spatial.distance.squareform : transforming condensed matrices into square ones.
  
      Examples
      --------
      >>> from scipy.cluster.hierarchy import single, cophenet
      >>> from scipy.spatial.distance import pdist, squareform
  
      Given a dataset ``X`` and a linkage matrix ``Z``, the cophenetic distance
      between two points of ``X`` is the distance between the largest two
      distinct clusters that each of the points:
  
      >>> X = [[0, 0], [0, 1], [1, 0],
      ...      [0, 4], [0, 3], [1, 4],
      ...      [4, 0], [3, 0], [4, 1],
      ...      [4, 4], [3, 4], [4, 3]]
  
      ``X`` corresponds to this dataset ::
  
          x x    x x
          x        x
  
          x        x
          x x    x x
  
      >>> Z = single(pdist(X))
      >>> Z
      array([[ 0.,  1.,  1.,  2.],
             [ 2., 12.,  1.,  3.],
             [ 3.,  4.,  1.,  2.],
             [ 5., 14.,  1.,  3.],
             [ 6.,  7.,  1.,  2.],
             [ 8., 16.,  1.,  3.],
             [ 9., 10.,  1.,  2.],
             [11., 18.,  1.,  3.],
             [13., 15.,  2.,  6.],
             [17., 20.,  2.,  9.],
             [19., 21.,  2., 12.]])
      >>> cophenet(Z)
      array([1., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 2., 2., 2., 2., 2.,
             2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 1., 2., 2.,
             2., 2., 2., 2., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2., 2.,
             1., 1., 2., 2., 2., 1., 2., 2., 2., 2., 2., 2., 1., 1., 1.])
  
      The output of the `scipy.cluster.hierarchy.cophenet` method is
      represented in condensed form. We can use
      `scipy.spatial.distance.squareform` to see the output as a
      regular matrix (where each element ``ij`` denotes the cophenetic distance
      between each ``i``, ``j`` pair of points in ``X``):
  
      >>> squareform(cophenet(Z))
      array([[0., 1., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
             [1., 0., 1., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
             [1., 1., 0., 2., 2., 2., 2., 2., 2., 2., 2., 2.],
             [2., 2., 2., 0., 1., 1., 2., 2., 2., 2., 2., 2.],
             [2., 2., 2., 1., 0., 1., 2., 2., 2., 2., 2., 2.],
             [2., 2., 2., 1., 1., 0., 2., 2., 2., 2., 2., 2.],
             [2., 2., 2., 2., 2., 2., 0., 1., 1., 2., 2., 2.],
             [2., 2., 2., 2., 2., 2., 1., 0., 1., 2., 2., 2.],
             [2., 2., 2., 2., 2., 2., 1., 1., 0., 2., 2., 2.],
             [2., 2., 2., 2., 2., 2., 2., 2., 2., 0., 1., 1.],
             [2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 0., 1.],
             [2., 2., 2., 2., 2., 2., 2., 2., 2., 1., 1., 0.]])
  
      In this example, the cophenetic distance between points on ``X`` that are
      very close (i.e., in the same corner) is 1. For other pairs of points is 2,
      because the points will be located in clusters at different
      corners - thus, the distance between these clusters will be larger.
  
      """
      Z = np.asarray(Z, order="c")
      is_valid_linkage(Z, throw=True, name="Z")
      Zs = Z.shape
      n = Zs[0] + 1
  
      zz = np.zeros((n * (n - 1)) // 2, dtype=np.double)
      # Since the C code does not support striding using strides.
      # The dimensions are used instead.
      Z = _convert_to_double(Z)
  
      _hierarchy.cophenetic_distances(Z, zz, int(n))
      if Y is None:
          return zz
  
      Y = np.asarray(Y, order="c")
      distance.is_valid_y(Y, throw=True, name="Y")
  
      z = zz.mean()
      y = Y.mean()
      Yy = Y - y
      Zz = zz - z
      numerator = Yy * Zz
      denomA = Yy**2
      denomB = Zz**2
      c = numerator.sum() / np.sqrt((denomA.sum() * denomB.sum()))
      return (c, zz)
  
imports: import warnings
  import bisect
  from collections import deque
  import numpy
  from .None import _hierarchy, _optimal_leaf_ordering
  import scipy.spatial.distance
  from scipy._lib._disjoint_set import DisjointSet
label_desc: Use image node_id 19 for calling a global function with example usage: cophenet(Z, Y) and returns: c,  zz, zz