Square-and-multiply reads the exponent as a bit string. The anchor exponent is small enough to inspect row by row.
Write the exponent in bits
The exponent 13 is written as bits 1101.
Read left to right
The square-and-multiply ladder reads those bits from left to right, so each row extends the value already built by earlier bits.
See what the bits mean
Bits 1101 represent 8 plus 4 plus 1, which is exponent 13.
One bit gives one row
There are 4 bits, so the ladder has 4 rows.
rows=4
Zero still gets a row
The zero bit does not disappear. It still causes a square-and-reduce row; it only skips the multiply by the base.
zero bit means square only
Summary
The exponent controls the row sequence; the table exposes that control instead of hiding it inside a giant decimal power.
exponent bits drive the ladder