Many options and I do have a recommendation (wait for it!). But being a discussion forum allow me to ramble..
So the task is to drop from 10V DC to 4.5V DC with a generous budget of 10 cents.
The key is to know how much power (Watts) is involved - hence my question about current required by the tree. Let's say with 30 LEDs we're talking 500 mA in round numbers. Power = Voltage x Current, so dropping 5.5V at 500mA is a power dissipation of about 3 Watts.
1. A resistor can drop the 5.5V. A 5 Watt resistor is common. The resistor value is Voltage / Current so 5.5V / 0.5A = 11 Ohms. Could be difficult to find an 11 Ohm, 5 Watt resistor. 10 Ohm and 12 Ohm are common values. Any variation/deviation of the current would proportionally change the voltage drop. Can probably find a 5 Watt resistor surplus for 10 cents...more like 50 cents (eBay).
2. A Zener diode can drop a fixed DC voltage. A common value is 5.6V which is close enough. A 5 Watt Zener is common. The voltage drop with variation/deviation in current is relatively constant. Not sure you can find one for 10 cents...more like 50 cents (eBay).
3. Not a single component but the venerable string-of-diodes can also drop a relatively fixed voltage. Using 2 cent diodes that drop about 0.7V, you could string together say 8 of them to drop about 5.6V. Note that your tree with a 4.5V 3-battery compartment probably operates fine with a fresh set of batteries of say 4.8V down to maybe 4V after some use. The point being the lack of resolution using this method is really not an issue as you can find some number of diodes to use which will put the dropped voltage in the operating voltage range of 4.0 to 4.8V (or whatever). Since many diodes are used, the power is spread out so low-cost generic 1 Amp diodes would be fine. That's maybe 20 cents in parts so probably the least expensive method but some assembly hassle.
4. Bottom line. My recommendation is to use an off-the-shelf DC-DC converter module for less than $1 (free shipping from Asia). Put the 10V DC in, adjust the module to 4.5V DC output. Variations/deviation in current will not affect the output voltage. I suppose if you're a kind person you could let it slide and call this "one" component since it's an assembled module. Additionally, because of the so-called switching technology used in this module, the 3 Watts of dissipated/wasted power of the other methods is reduced to less than 1 Watt which is a nice bonus.