I'm afraid I'm too lazy to write out by hand your entire matrix (please provide text rather than screenshots for data and code), but here's how you'd 'hack' it in to BioPython. If you can provide the raw data, it's not that difficult to parse as a plain file and convert it in to lists as needed below.

(The code in question is mostly here: https://github.com/biopython/biopython/blob/master/Bio/Phylo/TreeConstruction.py#L314)

from Bio import Phylo
from Bio.Phylo.TreeConstruction import DistanceTreeConstructor, DistanceMatrix

names = ['Aurora', 'Boylii', 'Cascadae']
values = [[0], [0.1, 0], [0.13, 0.7, 0]] #extend these lists as required for the whole matrix.

dm = DistanceMatrix(names=names, matrix=values)
constructor = DistanceTreeConstructor()

njtree = constructor.nj(dm) #change this line if you want something other than neighbour joining
print(njtree)

# Output:
# Tree(rooted=False)
#     Clade(branch_length=0, name='Inner1')
#         Clade(branch_length=-0.23499999999999996, name='Aurora')
#         Clade(branch_length=0.33499999999999996, name='Boylii')
#         Clade(branch_length=0.365, name='Cascadae')

You can then satisfy yourself that this is a tree:

Phylo.draw_ascii(njtree)
 , Aurora
 |
_|______________________________________________________________ Boylii
 |
 |____________________________________________________________________ Cascadae

I couldn't figure out a more elegant way to do this, though I'm sure there is one, but you can export this as a newick string like so:

print(njtree.format("newick"))
(Aurora:-0.23500,Boylii:0.33500,Cascadae:0.36500)Inner1:0.00000;

Just for fun I had ChatGPT try this too. Here's what it came up with.

A few notable differences:

• it uses numpy
• it is using a full matrix rather than triangular half matrix as biopython does
• It only produced a cladogram, so there are no branch lengths (I did tell the AI this, and it can produce updated code to change that) YMMV.

ChatGPT source:

import numpy as np

def create_newick_tree(dist_matrix, labels):
    """Create a Newick format tree from a distance matrix and labels."""
    num_taxa = len(labels)

    # Initialize the node labels
    node_labels = labels.copy()

    # Initialize the distance matrix
    dist = dist_matrix.copy()

    # Create the initial set of nodes
    nodes = [i for i in range(num_taxa)]

    # Keep track of the next available node ID
    next_node_id = num_taxa

    # Create the tree
    while len(nodes) > 1:
        # Find the two closest nodes
        min_dist = np.min(dist[np.triu_indices(num_taxa, k=1)])
        min_i, min_j = np.where(dist == min_dist)
        min_i, min_j = min_i[0], min_j[0]

        # Create a new node to represent the cluster
        new_node_id = next_node_id
        next_node_id += 1

        # Update the node labels and distances
        node_labels[new_node_id] = "(" + node_labels[min_i] + "," + node_labels[min_j] + ")"
        for k in range(num_taxa):
            if k != min_i and k != min_j:
                dist[min_i,k] = (dist[min_i,k] + dist[min_j,k]) / 2.0
                dist[k,min_i] = dist[min_i,k]
        dist = np.delete(dist, min_j, axis=0)
        dist = np.delete(dist, min_j, axis=1)
        node_labels = np.delete(node_labels, min_j)

        # Add the new node to the list of nodes
        nodes.append(new_node_id)

    # Return the Newick format string
    return node_labels[0] + ";"

# Example usage
dist_matrix = np.array([[0, 5, 9, 9], [5, 0, 10, 10], [9, 10, 0, 8], [9, 10, 8, 0]])
labels = np.array(["A", "B", "C", "D"])
newick_tree = create_newick_tree(dist_matrix, labels)
print(newick_tree)

It claims the result will be:

((A,B),(C,D));

DISCLAIMER I haven't confirmed that this is correct at anything other than a glance. Just include it for 'fun'.

I did challenge it to re-implement this task using biopython and it give a very similar solution to mine above.