Merkle Tree Proof Calculator
Bitcoin uses a binary Merkle tree to prove transaction inclusion. Proof size grows logarithmically with number of transactions.
Ethereum uses a Merkle-Patricia Trie to prove account/state inclusion. Proof size depends on state depth.
Enter values and click calculate to see proof sizes.
How It Works
Bitcoin Merkle proofs require ⌈log₂(N)⌉ hashes where N is the number of transactions.
Ethereum proofs typically involve 6-8 nodes for account state verification.
TL;DR
- Merkle Trees let Bitcoin verify a single transaction with just a few hashes.
- Ethereum upgrades the idea with a Merkle‑Patricia Trie to store accounts, contracts and receipts.
- Bitcoin’s proof size grows with log₂(num‑tx), Ethereum’s tries grow with state size.
- SPV wallets on Bitcoin rely on Merkle proofs; Ethereum dApps use trie proofs for state reads.
- Both systems cut bandwidth and storage, but Ethereum trades more computation for richer functionality.
When you hear the term Merkle Trees, you’re really hearing about the backbone that makes blockchains both trustworthy and lightweight. In Bitcoin, they enable a tiny mobile wallet to prove a payment without downloading the whole ledger. In Ethereum, they evolve into a Merkle‑Patricia Trie that can prove not just a payment but the entire state of a smart contract. This article walks through the math, the data structures, and the real‑world impact of these two approaches, ending with a side‑by‑side comparison you can reference when choosing a platform for your next project.
Merkle Tree is a binary hash tree where each leaf holds the hash of a data block and each parent stores the hash of its two children concatenated together. Invented by Ralph Merkle in 1988, the structure allows anyone to verify a single leaf by recomputing only the hashes along the path to the root. The root hash then serves as a compact commitment to the entire dataset. Bitcoin is a peer‑to‑peer digital cash system that records transactions in blocks chained together by proof‑of‑work. Bitcoin’s block header contains six fields, one of which is the Merkle root. Light clients-called Simple Payment Verification (SPV) nodes-download only these headers and a few proof elements to confirm that a transaction is included in a block. Simple Payment Verification is a method that allows a client to verify a transaction’s inclusion by checking a Merkle proof against the block’s Merkle root. The client needs three pieces of data: the transaction hash, the sibling hash at each tree level, and the block header’s Merkle root. With about log₂(N) hashes-where N is the number of transactions-validation is near‑instant. Block Header is a 80‑byte structure that records version, previous block hash, Merkle root, timestamp, difficulty target, and nonce. Because the header is small, SPV wallets can keep a copy of the longest chain without storing every transaction. Ethereum is a Turing‑complete blockchain platform that supports smart contracts, decentralized applications and a mutable global state. Ethereum replaces the simple Merkle tree with a Merkle‑Patricia Trie (MPT), a hybrid of a radix tree and a Merkle tree. The MPT can store arbitrary key‑value pairs while still providing cryptographic proof of any leaf. Merkle Patricia Trie is a data structure that combines a Merkle hash tree with a Patricia (compact prefix) trie, enabling efficient storage and verification of key‑value mappings. Ethereum maintains four distinct tries per block:- State Trie: maps each account address to its balance, nonce, storage root and code hash.
- Transaction Trie: stores each transaction’s RLP‑encoded data keyed by its index.
- Receipt Trie: holds transaction receipt data, never updated after block finalisation.
- Storage Tries: one per contract, storing the contract’s internal key‑value store.
Because the trie is keyed by the keccak256 hash of the path, any change-say, a balance update-only rewrites the nodes along that path, leaving the rest untouched. The new root hash becomes part of the block header, guaranteeing the integrity of the entire state.
Why Merkle Trees Matter for Blockchain Scalability
Both Bitcoin and Ethereum rely on Merkle‑based proofs to keep data transmission low. In a typical Bitcoin block with ~500 transactions, you need just 9 hashes (⌈log₂500⌉) to prove inclusion. Even a gigantic block with one million transactions requires only 20 hashes. For mobile wallets with megabytes of bandwidth, that reduction is decisive.
Ethereum’s MPT adds a layer of complexity. Proving that an account has a certain balance may involve traversing a path of around 6‑8 nodes, each containing a hash, a key fragment, and a value. The proof size is larger than Bitcoin’s, but it carries far more context-account nonce, storage root, and code hash-enabling dApps to read state without a full node.
Step‑by‑Step: Verifying a Bitcoin Transaction with SPV
- Obtain the transaction’s raw bytes and compute its double‑SHA256 hash.
- Request from a full node the list of sibling hashes that sit next to your hash at each tree level.
- Iteratively concatenate your hash with its sibling (left‑to‑right order), hash the pair, and repeat until you reach the root.
- Compare the resulting root with the Merkle root stored in the block header you already trust.
If the numbers match, the transaction is undeniably part of that block. No need to download the other 499 transactions.
Step‑by‑Step: Proving an Ethereum Account Balance
- Take the account address, hash it with keccak256, and split it into 32‑byte nibbles.
- Follow the nibbles down the State Trie, collecting the hash of each visited node.
- When you reach the leaf, you’ll see the RLP‑encoded account data (balance, nonce, storageRoot, codeHash).
- Combine the collected node hashes to reconstruct the state root. Verify that this root matches the one in the block header.
This proof can be sent to any light client, enabling it to read balances, token holdings or contract storage without syncing the whole chain.
Bitcoin vs Ethereum: Direct Comparison
| Feature | Bitcoin Merkle Tree | Ethereum Merkle‑Patricia Trie |
|---|---|---|
| Primary purpose | Transaction inclusion proof | Full state & contract storage verification |
| Data structure type | Binary hash tree | Patricia trie + Merkle hashing |
| Proof size (average block) | ≈9 hashes (500 tx) | 6‑8 nodes per account proof |
| Impact on light clients | Enables SPV wallets | Enables state‑reading dApps & light sync |
| Computational cost | Low (hash concat) | Higher (RLP decoding, trie traversal) |
| Storage requirement | Only block headers (~80bytes each) | Headers + trie nodes (≈1‑2GB for full sync) |
| Typical use‑case | Payment verification | Smart contract state queries, airdrop proofs |
Real‑World Applications Beyond Payments
Developers love Merkle proofs for airdrops. By storing a single root hash on‑chain, an issuer can let each recipient generate a proof off‑chain and claim tokens without the contract having to store every address. JavaScript libraries like merkletreejs work with keccak256 to build those trees in minutes.
Git, BitTorrent and IPFS also rely on Merkle structures to guarantee file integrity across distributed peers. The same guarantees that make a Bitcoin transaction tamper‑proof also protect a software repository’s history.
Future Directions: Verkle Trees and Beyond
Researchers are already eyeing Verkle trees-a vector‑commitment variant that can shrink proof sizes from dozens of kilobytes to a few hundred bytes. For Ethereum, this could mean faster sync times and cheaper on‑chain verification, while Bitcoin might see smaller Merkle proofs for future scaling solutions like Bitcoin Taproot’s script path spends.
As transaction volumes climb and smart contracts become more sophisticated, the pressure to keep proofs tiny while preserving security will only increase. The core idea remains the same: a succinct cryptographic commitment that anyone can verify with minimal data.
Frequently Asked Questions
What is a Merkle proof?
A Merkle proof is a list of sibling hashes that, when combined with the target leaf hash, reproduces the Merkle root stored in a block header. Matching the computed root with the on‑chain root proves that the leaf belongs to the dataset.
Why does Ethereum use a Patricia trie instead of a plain Merkle tree?
A Patricia trie compresses long paths of empty branches, reducing the number of nodes needed to store sparse key‑value maps like account balances. Combining this with Merkle hashing gives cryptographic guarantees while keeping storage reasonable for a state that can hold millions of accounts.
Can I use Merkle proofs on a mobile wallet?
Yes. Light Bitcoin wallets (SPV) rely on Merkle proofs to verify payments without downloading the whole blockchain. Ethereum light clients use similar trie proofs to read contract state.
How large is a typical Merkle proof in Bitcoin?
For a block with 500 transactions, a proof contains about nine 32‑byte hashes, roughly 288bytes total. Even a block with 1million transactions needs only around 20 hashes.
What tools help me build Merkle trees for airdrops?
JavaScript libraries like merkletreejs paired with keccak256 let you create leaf nodes from encoded address‑balance pairs, generate the root hash for on‑chain storage, and export individual proofs for users to claim tokens.
Author
Ronan Caverly
I'm a blockchain analyst and market strategist bridging crypto and equities. I research protocols, decode tokenomics, and track exchange flows to spot risk and opportunity. I invest privately and advise fintech teams on go-to-market and compliance-aware growth. I also publish weekly insights to help retail and funds navigate digital asset cycles.