<｜begin▁of▁sentence｜>### Instruction:
Given a binary tree with 127 nodes arranged as a perfect binary tree (where every internal node has exactly two children and all leaf nodes are at the same level), how can you use CairoDendrogramDrawer in igraph to draw the dendrogram of its community structure, styled with a color gradient from red through green to blue, ensuring each node is represented by a square cell in the matrix with printed values representing the community IDs, with a grid width of 0.5 and a border width of 2, then save this visualization as 'dendrogram.png'?
### Response:
import igraph as ig
from igraph.drawing import Plot
from igraph.drawing.colors import AdvancedGradientPalette

# Generate a random graph and corresponding dendrogram
graph = ig.Graph.Tree(127, 2)
dendrogram = graph.community_edge_betweenness().as_clustering().to_dendrogram()

# Create a plot with 600x600 pixels
plot = Plot(bbox=(600, 600), background='white')

# Create a gradient palette for styling
palette = AdvancedGradientPalette(['red', 'green', 'blue'], n=127)

# Add the dendrogram to the plot
plot.add(
    ig.CairoDendrogramDrawer(dendrogram, plot.bbox),
    row_names=[str(i) for i in range(graph.vcount())],
    col_names=[str(i) for i in range(graph.vcount())],
    style='palette',
    palette=palette,
    square=True,
    grid_width=0.5,
    border_width=2,
    values=True,
    value_format='%d'
)

# Save the plot
plot.save('dendrogram.png')
<|EOT|>
