This may not be an exact answer but some information of interest related to your question
Other answers have mentioned various ways in which the binary data structure can be represented and you might want to use one of them but mostly when using databases the Adjacency List Model and Nested Set Model are used for representing Binary Trees.
Binary Trees are mostly not used for storing data on disk and some alternatives are usually preferred.If you are interested in optimized data access, search,insertion and deletion on disk then you may want to consider these alternatives
B-Tree data structure is one of the alternatives used for storage on disk.It provides search, sequential access, insert, and delete operations in logarithmic time and is optimized for disk storage.
B-Tree data structure was created by Rudolf Bayer and Ed McCreight in 1972 to overcome a shortfall of Binary Tree i.e. using too many disk reads.B-Tree reduces the number of disk reads by allowing more keys in a single node ( typically making a B-tree node as large as the sector minimizes the number of access times).
SQL Server uses a further optimized extension of B-Tree data structure known as B+Tree in which only the leaf nodes hold the actual data and the rest of the nodes are pointers.
According to Wikipedia B+Trees are also used for storing directory structures by some filesystems:
The ReiserFS, NSS, XFS, JFS, ReFS, and BFS filesystems all use this
type of tree for metadata indexing; BFS also uses B+ trees for storing
directories. NTFS uses B+ trees for directory and security-related
How will a tree be represented as bits on the disk?
Trees can be represented by using static data structures like arrays in a fashion similar to this,consider each row tag as an array
//Node A can be represented as
[row 1][node A data in bits][separator bits]
[pointer to B i.e row 2][separator bits]
[pointer to C i.e. row 3][row1end]
//Node B can be represented as
[row 2][node B data in bits] [separator bits]
[no child flag][separator bits][no child flag][row2end]
//Node C can be represented as
[row 3][node C data in bits] [separator bits]
[no child flag][separator bits][no child flag][row3end]
I'd recommend studying some file formats so that you know how to represent logical data in a file, this also might be helpful