<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def dendrogram(df, method='average', filter=None, n=0, p=0, sort=None, orientation=None, figsize=None, fontsize=16, inline=False ):
 """ Fits a `scipy` hierarchical clustering algorithm to the given DataFrame's variables and visualizes the results as a `scipy` dendrogram. The default vertical display will fit up to 50 columns. If more than 50 columns are specified and orientation is left unspecified the dendrogram will automatically swap to a horizontal display to fit the additional variables. :param df: The DataFrame whose completeness is being dendrogrammed. :param method: The distance measure being used for clustering. This is a parameter that is passed to `scipy.hierarchy`. :param filter: The filter to apply to the heatmap. Should be one of "top", "bottom", or None (default). :param n: The cap on the number of columns to include in the filtered DataFrame. :param p: The cap on the percentage fill of the columns in the filtered DataFrame. :param sort: The sort to apply to the heatmap. Should be one of "ascending", "descending", or None. :param figsize: The size of the figure to display. This is a `matplotlib` parameter which defaults to `(25, 10)`. :param fontsize: The figure's font size. :param orientation: The way the dendrogram is oriented. Defaults to top-down if there are less than or equal to 50 columns and left-right if there are more. :param inline: Whether or not the figure is inline. If it's not then instead of getting plotted, this method will return its figure. :return: If `inline` is False, the underlying `matplotlib.figure` object. Else, nothing. """

    if not figsize:
        if len(df.columns) <= 50 or orientation == 'top' or orientation == 'bottom':
            figsize = (25, 10)
        else:
            figsize = (25, (25 + len(df.columns) - 50)*0.5)
    
    plt.figure(figsize=figsize)
    gs = gridspec.GridSpec(1, 1)
    ax0 = plt.subplot(gs[0])

    df = nullity_filter(df, filter=filter, n=n, p=p)
    df = nullity_sort(df, sort=sort)

    # Link the hierarchical output matrix, figure out orientation, construct base dendrogram.
    x = np.transpose(df.isnull().astype(int).values)
    z = hierarchy.linkage(x, method)

    if not orientation:
        if len(df.columns) > 50:
            orientation = 'left'
        else:
            orientation = 'bottom'

    hierarchy.dendrogram(z,
                         orientation=orientation,
                         labels=df.columns.tolist(),
                         distance_sort='descending',
                         link_color_func=lambda c: 'black',
                         leaf_font_size=fontsize,
                         ax=ax0
                        )

    # Remove extraneous default visual elements.
    ax0.set_aspect('auto')
    ax0.grid(b=False)
    if orientation == 'bottom':
        ax0.xaxis.tick_top()
    ax0.xaxis.set_ticks_position('none')
    ax0.yaxis.set_ticks_position('none')
    ax0.spines['top'].set_visible(False)
    ax0.spines['right'].set_visible(False)
    ax0.spines['bottom'].set_visible(False)
    ax0.spines['left'].set_visible(False)
    ax0.patch.set_visible(False)

    # Set up the categorical axis labels and draw.
    if orientation == 'bottom':
        ax0.set_xticklabels(ax0.xaxis.get_majorticklabels(), rotation=45, ha='left')
    elif orientation == 'top':
        ax0.set_xticklabels(ax0.xaxis.get_majorticklabels(), rotation=45, ha='right')
    if orientation == 'bottom' or orientation == 'top':
        ax0.tick_params(axis='y', labelsize=int(fontsize / 16 * 20))
    else:
        ax0.tick_params(axis='x', labelsize=int(fontsize / 16 * 20))

    if inline:
        plt.show()
    else:
        return ax0